Name: __________________________________________________
Big Bend Community College
Emporium Model
Math Courses
Workbook
A workbook to supplement
video lectures and online homework by:
Tyler Wallace
Salah Abed
Sarah Adams
Mariah Helvy
April Mayer
Michele Sherwood
1
This project was made possible in part by a federal STEM-HSI grant under Title III part F and by the generous
support of Big Bend Community College and the Math Department.
Copyright 2012, Some Rights Reserved CC-BY-NC-SA. This work is a combination of original work and a
derivative of Prealgebra Workbook, Beginning Algebra Workbook, and Intermediate Algebra Workbook by
Tyler Wallace which all hold a CC-BY License. Cover art by Sarah Adams with CC-BY-NC-SA license.
Emporium Model Math Courses Workbook by Wallace, Abed, Adams, Helvy, Mayer,
Sherwood is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike
3.0 Unported License (http://creativecommons.org/licenses/by-nc-sa/3.0/)
You are free:
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To share: To copy, distribute and transmit the work
To Remix: To adapt the work
Under the following conditions:
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Attribution: You must attribute the work in the manner specified by the authors or licensor (but not in any
way that suggests that they endorse you or your use of the work).
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Share Alike: If you alter, transform, or build upon this work, you may distribute the resulting work only
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status is in no way affected by the license.
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2
Conversion Factors
LENGTH
English
Metric (meter)
12 in = 1 ft
3 ft = 1 yd
1 mi = 5280 ft
1000 mm = 1 m
100 cm = 1 m
10 dm = 1 m
1 dam = 10 m
1 hm = 100 m
1 km = 1000 m
TEMPERATURE
English to Metric
1 in = 2.54 cm
VOLUME
English
Metric (liter)
8 fl oz = 1 cup (c)
2 cups (c) = 1 pint (pt)
2 pints (pt) = 1 quart (qt)
4 quarts (qt) = 1 gallon (gal)
1000 mL = 1 L
100 cL = 1 L
10 dL = 1 L
1 daL = 10 L
1 hL = 100 L
1 kL = 1000 L
1 mL = 1 cc = 1 cm3
TIME
60 seconds (sec) = 1 minute (min)
60 minutes (min) = 1 hour (hr)
24 hours (hr) = 1 day
52 weeks = 1 year
365 days = 1 year
English to Metric
1 gallon (gal) = 3.79 liter (L)
1 in3 = 16.39 mL
WEIGHT (MASS)
English
Metric (gram)
16 oz = 1 pound (lb)
2,000 lb = 1 Ton (T)
1000 mg = 1 g
100 cg = 1 g
10 dg = 1 g
1 dag = 10 g
1 hg = 100 g
1 kg = 1000 g
INTEREST
.
Simple:
Continuous:
A = Pert
Compound:
Annual: n=1
Semiannual: n=2
English to Metric
Quarterly: n=4
2.20 lb = 1 kg
3
.
Geometric Formulas
Name
Rectangle
Diagram
Area
w
A = lw
l
h
Parallelogram
A = bh
b
Triangle
h
b
a
h
Trapezoid
P = 2l + 2w
A=
bh
A=
h(a + b)
b
d
Circle
Name
r
A = πr2
Diagram
C = πd = 2πr
Volume
Rectangular
Solid
h
w
l
V = lwh
r
Right Circular
Cylinder
V = πr2h
h
Right Circular
Cone
h
V=
πr2h
V=
πr3
r
Sphere
r
Right Triangle
Pythagorean Theorem:
a2
+
b2
c2
=
4
c
a
b
Table of Contents
Unit 1: Integers and Algebraic Expressions .................................................................................... 8
1.1 Whole Numbers and Decimals ..................................................................................... 9
1.2 Operations with Integers ............................................................................................ 19
1.3 Exponents and Order of Operations ........................................................................... 25
1.4 Simplify Algebraic Expressions .................................................................................... 29
Unit 2: Fractions ............................................................................................................................ 35
2.1 Prime Factorization ..................................................................................................... 36
2.2 Reduce Fractions ......................................................................................................... 39
2.3 Multiply and Divide Fractions ..................................................................................... 45
2.4 Least Common Multiple .............................................................................................. 53
2.5 Add and Subtract Fractions......................................................................................... 57
2.6 Order of Operations with Fractions ............................................................................ 60
2.7 Mixed Numbers........................................................................................................... 63
Unit 3: Linear Equations................................................................................................................ 66
3.1 One Step Equations ..................................................................................................... 67
3.2 Two Step Equations .................................................................................................... 71
3.3 General Linear Equations ............................................................................................ 73
3.4 Equations with Decimals and Fractions ...................................................................... 76
Unit 4: Stats, Graphing, Proportions, and Percents ...................................................................... 78
4.1 Averages ...................................................................................................................... 79
4.2 Probability and Plotting Points ................................................................................... 84
4.3 Rates and Unit Rates ................................................................................................... 88
4.4 Proportions and Applications ..................................................................................... 90
4.5 Introduction to Percents ............................................................................................. 92
4.6 Translate Percents and Applications .......................................................................... 94
4.7 Percents as Proportions and Applications .................................................................. 99
Unit 5: Geometry and Intro to Polynomials................................................................................ 102
Conversion Factors.......................................................................................................... 103
5.1 Convert Units ............................................................................................................ 104
Geometry Formulas ........................................................................................................ 107
5.2 Area, Volume, and Temperature .............................................................................. 108
5.3 Pythagorean Theorem .............................................................................................. 119
5.4 Introduction to Polynomials ..................................................................................... 122
5.5 Scientific Notation..................................................................................................... 126
Unit 6: Proficiency Exam #1 ........................................................................................................ 128
5
Unit 7: Linear Equations and Applications .................................................................................. 129
7. 1 Order of Operations ................................................................................................. 130
7.2 Evaluate and Simplify Algebraic Expressions ............................................................ 134
7.3 Solve Linear Equations .............................................................................................. 138
7.4 Formulas.................................................................................................................... 142
7.5 Absolute Value Equations ......................................................................................... 146
7.6 Word Problems ......................................................................................................... 149
7.7 Age Problems ............................................................................................................ 154
7.8 Distance, Rate, Time Problems ................................................................................. 157
7.9 Inequalities ................................................................................................................ 160
Unit 8: Graphing Linear Equations .............................................................................................. 165
8.1 Slope.......................................................................................................................... 166
8.2 Equations of Lines ..................................................................................................... 167
8.3 Parallel and Perpendicular Lines ............................................................................... 174
Unit 9: Polynomials ..................................................................................................................... 177
9.1 Exponents.................................................................................................................. 178
9.2 Scientific Notation..................................................................................................... 184
9.3 Add, Subtract, Multiply Polynomials ........................................................................ 189
9.4 Polynomial Long Division .......................................................................................... 197
Unit 10: Factoring ....................................................................................................................... 201
10.1 Factor Common Factors and Grouping ................................................................... 202
10.2 Factor Trinomials ................................................................................................... 207
10.3 Factoring Tricks ....................................................................................................... 211
10.4 Factoring Strategy ................................................................................................... 218
10.5 Solving Equations by Factoring ............................................................................... 219
Unit 11: Rational Expressions ..................................................................................................... 224
11.1 Reduce Rational Expressions .................................................................................. 225
11.2 Multiply and Divide Rational Expressions ............................................................... 228
11.3 Add and Subtract Rational Expressions .................................................................. 231
Conversion Factors.......................................................................................................... 237
11.4 Dimensional Analysis .............................................................................................. 238
Unit 12: Proficiency Exam #2 ...................................................................................................... 241
6
Unit 13: Compound Inequalities ................................................................................................. 242
13.1 Review Inequalities ................................................................................................. 243
13.2 Compound Inequalities ........................................................................................... 245
13.3 Absolute Value Inequalities .................................................................................... 250
Unit 14: Systems of Equations .................................................................................................... 253
14.1 Systems by Graphing............................................................................................... 254
14.2 Systems by Substitution and Addition/Elimination ................................................ 256
14.3 Systems with Three Variables ................................................................................. 270
14.4 Applications of Systems .......................................................................................... 274
Unit 15: Radicals ......................................................................................................................... 286
15.1 Simplify Radicals ..................................................................................................... 287
15.2 Add, Subtract, and Multiply Radicals ...................................................................... 291
15.3 Rationalize Denominators....................................................................................... 298
15.4 Rational Exponents ................................................................................................. 302
15.5 Radicals of Mixed Index .......................................................................................... 306
15.6 Complex Numbers................................................................................................... 309
Unit 16: Quadratics, Rational Equations, and Applications ........................................................ 315
16.1 Complete the Square .............................................................................................. 316
16.2 Quadratic Formula .................................................................................................. 321
16.3 Equations with Radicals .......................................................................................... 325
16.4 Equations with Exponents ...................................................................................... 329
16.5 Rectangle Problems ................................................................................................ 333
16.6 Rational Equations .................................................................................................. 338
16.7 Work Problems ....................................................................................................... 341
16.8 Distance and Revenue Problems ............................................................................ 343
16.9 Compound Fractions ............................................................................................... 350
Unit 17: Functions ....................................................................................................................... 354
17.1 Evaluate Functions .................................................................................................. 355
17.2 Operations on Functions ......................................................................................... 359
17.3 Inverse Functions .................................................................................................... 367
17.4 Graphs of Quadratic Functions ............................................................................... 372
17.5 Exponential Equations ............................................................................................ 374
17.6 Compound Interest ................................................................................................. 377
17.7 Logarithms .............................................................................................................. 379
Unit 18: Proficiency Exam #3 ...................................................................................................... 382
7
Unit 1:
Integers and Algebraic Expressions
To work through the unit you should:
1.
2.
3.
4.
5.
6.
7.
Watch a video, as you watch, fill out the workbook (top and example sections).
Complete Q1 and Q2 in WAMAP, put your work in the bottom sections of the page.
Repeat #1 and #2 with each page until you reach
.
Complete the homework assignment on your own paper.
Repeat #1-#4 until you reach the end of the unit.
Complete the practice test on your own paper.
Take the unit exam.
8
1.1a Whole Numbers and Decimals – Rounding Whole Numbers
Whole numbers are:
Place Value:
4,
2
8
7,
1
9
2
Rounding: Look at the _____________ digit. Round up if it is __________ and round down if it is __________
Example 1:
Example 2:
Round 5,459,246
To the nearest thousand
Round 5,459,246
To the nearest hundred-thousand
Q 1:
Q 2:
9
1.1b Whole Numbers and Decimals – Rounding Decimals
Decimals are ___________ of the ______________
Place Value:
8.
Example 1:
1
7
2
6
3
Example 2:
Round 4.01276
To the nearest thousandth
Q 1:
9
Round 4.01276
To the nearest hundredth
Q 2:
10
1.1c Whole Numbers and Decimals – Add Whole Numbers
To add we _________________ place values and work __________ to _____________
Example 1:
Example 2:
458 + 321
Q 1:
716 + 485
Q 2:
11
1.1d Whole Numbers and Decimals – Add Decimals
When adding decimals we must _________________ the ___________________
Place decimal:
Example 1:
Example 2:
4.21 + 8.962
Q 1:
0.523 + 0.08
Q 2:
12
1.1e Whole Numbers and Decimals – Subtract Whole Numbers
To subtract we __________________ place value and work ___________ to ______________
Example 1:
Example 2:
967 − 341
Q 1:
5037 − 2419
Q 2:
13
1.1f Whole Numbers and Decimals – Subtract Decimals
When subtracting decimals we must _________________ the ___________________
Important: We may need additional ________________ to line up!
Place decimal:
Example 1:
Example 2:
3.4 − 1.29
Q 1:
4.03 − 0.051
Q 2:
14
1.1g Whole Numbers and Decimals – Multiply Whole Numbers
Multiply __________ the digits together
Use _____ to hold place value
After multiplying we _________
Different ways show “multiply”:
Example 1:
Example 2:
167(48)
23 ∙ 56
Q 1:
Q 2:
15
1.1h Whole Numbers and Decimals – Multiply Decimals
Place decimal:
Example 1:
Example 2:
2.6(3.52)
4.2 ∙ 1.8
Q 1:
Q 2:
16
1.1i Whole Numbers and Decimals – Divide Whole Numbers
Long division places the ____________ number in front!
The leftovers:
Different ways to show “divide”:
Example 1:
Example 2:
12024
24
452 ÷ 13
Q 1:
Q 2:
17
1.1j Whole Numbers and Decimals – Divide Decimals
No decimals in the ________________ or ________________
Move the ________________ in both the _______________ and _____________________
If you run out of digits you can _______________________
Place decimal:
Example 1:
Example 2:
2.568
2.4
Q 1:
19.5 ÷ 25
Q 2:
You have completed the videos for 1.1 Whole Numbers and Decimals. On your own paper complete
the homework assignment.
18
1.2a Operations with Integers – Integers and Absolute Value
Integers:
Opposite means a _________________ over ____________________ on the ________________________
The symbol ____ means _________________, __________________ AND __________________ !
Absolute Value is the _______________ from zero. It is always ________________
Example 1:
Example 2:
−(−6)
Example 3:
−(3)
Example 4:
|−8|
Example 5:
|4|
Example 6:
−| − 7|
−|2|
Q 1:
Q 2:
Q 3:
Q 4:
Q 5:
Q 6:
19
1.2b Operations with Integers – Multiply and Divide with Different Signs
A pattern to multiplying:
2∙2=
2∙1=
2∙0=
2 ∙ (−1) =
Multiplying and dividing with a negative and a positive (or positive and negative) is a _________________
To remember: When _________ things happen to _________ people it is a _________ thing
To remember: When _________ things happen to _________ people it is a _________ thing
Example 1:
Example 2:
3(−8)
−54 ÷ 9
Q 1:
Q 2:
20
1.2c Operations with Integers – Multiply and Divide with the Same Sign
A pattern to multiplying:
−2 ∙ 2 =
−2 ∙ 1 =
−2 ∙ 0 =
−2 ∙ (−1) =
Multiplying and dividing a negative times a negative is a ________________
To remember: When _________ things happen to _________ people it is a _________ thing
Example 1:
Example 2:
−15
−3
−7(−4)
Q 1:
Q 2:
21
1.2d Operations with Integers – Add with the Same Sign
Adding and Subtracting Integers: Keep the ___________ with the ______________ after it
Double signs: 3 + (−4) =
3 − (−4) =
Visualize: −2 + (−3):
Add with the same sign:
Example 1:
Example 2:
(−6) + (−8)
−7 + (−4)
Q 1:
Q 2:
22
1.2e Operations with Integers – Add with Different Signs
Visualize: −2 + 4
Visualize: 1 + (−3)
Adding with different signs:
Example 1:
Example 2:
5 + (−2)
Q 1:
−9 − (−4)
Q 2:
23
1.2f Operations with Integers – Add and Subtract Several Integers
When adding and subtracting many integers we work __________ to ____________
Example 1:
Example 2:
−5 + (−2) − (−6) − 4 + 8
Q 1:
4 − 8 + (−3) − (−1) + 3
Q 2:
You have completed the videos for 1.2 Operations with Integers. On your own paper complete the
homework assignment.
24
1.3a Exponents and Order of Operations – Exponents
Exponents are ______________________________
53 =
Example 1:
Example 2:
72
25
Q 1:
Q 2:
25
1.3b Exponents and Order of Operations – Exponents on Negatives
Exponents only effect what they are ________________.
(−5)2 =
−52 =
Example 1:
Example 2:
−34
Q 1:
(−2)6
Q 2:
26
1.3c Exponents and Order of Operations – Order of Operations
Why we need an order:
2 + 3(4)
2 + 3(4)
The order:
Example 1:
Example 2:
3
24 ÷ 6 ∙ 2 − 32 (7 − 9)
2 + 5(4 − 7)
Q 1:
Q 2:
27
1.3d Exponents and Order of Operations – Order with Absolute Value
Absolute values works just like ______________ but makes the number inside __________
after it has been _______________
Example 1:
Example 2:
2
|4(2) − 6|3 − 42
3 − 2|7 − 4 |
Q 1:
Q 2:
You have completed the videos for 1.3 Exponents and Order of Operations. On your own paper
complete the homework assignment.
28
1.4a Simplify Algebraic Expressions – Substitute a Value
I have a ______________ eggs means I have _____ eggs.
Variables are ____________ that represent ______________ amounts
If we know the amount we can ____________ it in an expression
Whenever we make a substitution or __________________ put it in ____________________
Example 1:
Example 2:
2
Evaluate: 𝑏 2 − 4𝑎𝑐
When 𝑎 = 2, 𝑏 = −3, 𝑎𝑛𝑑 𝑐 = −5
Evaluate: −𝑥 − 7𝑥 − 12
When 𝑥 = −4
Q 1:
Q 2:
29
1.4b Simplify Algebraic Expressions – Is it a Solution?
An equation is made up of two ______________ expressions
A solution is the value of the ______________ that makes the equation ________________
Example 1:
Example 2:
Is 𝑥 = 3 the solution to
−2𝑥 + 7 = 1 ?
Q 1:
Is 𝑥 = −3 the solution to
2𝑥 − 5 = 7𝑥 + 5?
Q 2:
30
1.4c Simplify Algebraic Expressions – Combine Like Terms
John has 5 cats and 3 dogs. Sue has 2 cats and 1 dog. Together they have ___ cats and ___ dogs.
Terms are __________ and ____________ that are ________________ together
Like terms are terms that have matching _______________ and _______________
Combine like terms: ___________ the coefficients or ______________ from __________________
Example 1:
Example 2:
5𝑥 2 − 2𝑥 − 9 + 4𝑥 − 7𝑥 2 + 6
9𝑥 + 2𝑦 − 7𝑥 − 5𝑦 + 2𝑥
Q 1:
Q 2:
31
1.4d Simplify Algebraic Expressions – Distributive Property
Multiplication is __________________
3(2𝑥 + 5) =
Distributive Property: 3(2𝑥 + 5)
Example 1:
Example 2:
7(9𝑥 2 − 7𝑥 + 8)
−4(2𝑥 + 5𝑦 − 7)
Q 1:
Q 2:
32
1.4e Simplify Algebraic Expressions – Distribute and Combine Like Terms
Order of operations states we ________________ before we ____________________
Therefore we will _________________ first and then _________________________ second
Example 1:
Example 2:
2(4𝑥 2 − 6𝑥 + 1) − (𝑥 2 + 5𝑥 + 3)
5(2𝑥 + 6𝑦 − 2) − 4(𝑥 + 3 − 6𝑦)
Q 1:
Q 2:
33
1.4f Simplify Algebraic Expressions – Perimeter Problems
Perimeter:
Find the perimeter by ________________ the sides together
Example 1:
Example 2:
7 − 2𝑥
8 − 4𝑥
3𝑥 + 7
5𝑥 − 4
−9 − 2𝑥
3+𝑥
3
6 − 4𝑥
1−𝑥
Q 1:
Q 2:
You have completed the videos for 1.4 Simplify Algebraic Expressions. On your own paper complete
the homework assignment.
Congratulations! You made it through the material for Unit 1: Integers and Algebraic Expressions. It
is time to prepare for your exam. On a separate sheet of paper complete the practice test. Once you
have completed the practice test, ask your instructor to take the test. Good luck!
34
Unit 2:
Fractions
To work through the unit you should:
1.
2.
3.
4.
5.
6.
7.
Watch a video, as you watch, fill out the workbook (top and example sections).
Complete Q1 and Q2 in WAMAP, put your work in the bottom sections of the page.
Repeat #1 and #2 with each page until you reach
.
Complete the homework assignment on your own paper.
Repeat #1-#4 until you reach the end of the unit.
Complete the practice test on your own paper.
Take the unit exam.
35
2.1a Prime Factorization – Prime and Composite
Prime numbers are divisible by _____ and ___________
Examples of Primes:
Composite numbers are divisible by ______________________
Example 1:
Example 2:
Prime or Composite:
89
Q 1:
Prime or Composite:
147
Q 2:
36
2.1b Prime Factorization – Divisibility Tests
A number is divisible by a smaller number if the small number ___________________ into the number
Divisibility tests:
2:
3:
5:
7, 11, 13, 17, 19:
Example 1:
2730 is divisible by which prime numbers?
Example 2:
133 is divisible by which numbers?
Q 1:
Q 2:
37
2.1c Prime Factorization – Prime Factorization
Prime Factorization: a _______________ of __________________ numbers
To find a prime factorization we divide by _________________
Example 1:
Example 2:
Find the prime factorization of
360
Q 1:
Find the prime factorization of
1224
Q 2:
You have completed the videos for 2.1 Prime Factorization. On your own paper complete the
homework assignment.
38
2.2a Reduce Fractions – Introduction to Fractions
Fraction is a ______________ of a ________________
4
Example: 5 where the 4 is the ________, called the _______________ and the 5 is the ________ called the
____________
Example 1:
Example 2:
What fraction is shaded?
Q 1:
What fraction is shaded?
Q 2:
39
2.2b Reduce Fractions – Convert Fractions to Decimals
The fraction bar represents _______________________
To convert a fraction to a decimal we _________________
To help with this we will use a ____________________
If the decimal repeats we will use a ___________
Example 1:
Example 2:
Convert to decimal:
Q 1:
7
32
Convert to decimal: 99
32
Q 2:
40
2.2c Reduce Fractions – Equivalent Fractions
Equivalent fractions:
To find an equivalent fraction ______________________________ the _____________________________
and __________________________ by the ______________________________
Example 1:
Find three equivalent fractions:
3
7
Q 1:
Example 2:
Find three equivalent fractions:
4
3
Q 2:
41
2.2d Reduce Fractions – Reduce with Prime Factorizations
Reduced Fraction: The __________________ and ___________________ have no common _____________
To reduce we find the ___________________________ and divide out ________________________
Example 1:
Example 2:
24
36
Q 1:
105
70
Q 2:
42
2.2e Reduce Fractions - Reduce
Sometimes we can _________ the common factors and ___________________________
Example 1:
Example 2:
24
36
Q 1:
105
70
Q 2:
43
2.2f Reduce Fractions – Reduce with Variables
When reducing with variables, ___________________ the variables that are in _______________
With exponents it may help to _____________________
Example 1:
Example 2:
2
27𝑎3 𝑏𝑐
9𝑎2 𝑏 2 𝑐
4𝑥 𝑦𝑧
10𝑥𝑦 3
Q 1:
Q 2:
You have completed the videos for 2.2 Reduce Fractions. On your own paper complete the
homework assignment.
44
2.3a Multiply and Divide Fractions – Multiply with No Reducing
Multiply the ____________________ together and multiply the _______________________ together
Example 1:
Example 2:
4 5
∙
7 3
Q 1:
1 5
∙
6 4
Q 2:
45
2.3b Multiply and Divide Fractions – Multiply with Reducing
Consider:
4 6
∙ =
9 5
But if we factor each:
4 6
∙ =
9 5
When _____________ we can _____________ a common factor from the ____________ and ____________
Example 1:
Example 2:
6 14
∙
35 15
Q 1:
6 35
∙
14 13
Q 2:
46
2.3c Multiply and Divide Fractions – Multiply with Variables
With exponents on variables it may help to _______________________
Remember, repeated ________________ is done with ____________________
Example 1:
Example 2:
2
30𝑎 21𝑎𝑏
∙
3𝑏 2 10
6𝑥 𝑦 14𝑦
∙
7
3𝑥
Q 1:
Q 2:
47
2.3d Multiply and Divide Fractions – Multiply with Whole Numbers and Fractions
Whole numbers can be made into fractions by putting them over _____
Example 1:
Example 2:
3
∙ 20
8
Q 1:
35 ∙
Q 2:
48
6
7
2.3e Multiply and Divide Fractions - Reciprocals
Reciprocal:
Reciprocals multiply to ____
Example 1:
Example 2:
6
Find the reciprocal of −8
Find the reciprocal of 5
Q 1:
Q 2:
49
2.3f Multiply and Divide Fractions – Divide Fractions
Divide fractions by _________________ by the __________________
Example 1:
Example 2:
3
6
÷
10 15
14 35
÷
15 6
Q 1:
Q 2:
50
2.3g Multiply and Divide Fractions – Divide with Variables
With exponents on variables it may help to _______________________
Remember, repeated ________________ is done with ____________________
Example 1:
Example 2:
10𝑥
10
÷
2
3𝑦
21𝑥𝑦
Q 1:
14𝑚
7
÷
3𝑛
6𝑚2 𝑛
Q 2:
51
2.3h Multiply and Divide Fractions – Divide with Whole Numbers and Fractions
Whole numbers can be made into fractions by putting them over _____
Example 1:
Example 2:
7
28 ÷
8
Q 1:
4
÷ 14
9
Q 2:
You have completed the videos for 2.3 Multiply and Divide Fractions. On your own paper complete
the homework assignment.
52
2.4a Least Common Multiple - Multiples
Multiples are the found by ______________ by other numbers
Example 1:
Find the first three multiples of 8
Example 2:
Find the first three multiples of −7
Q 1:
Q 2:
53
2.4b Least Common Multiple – LCM Using Mental Math
Least Common Multiple (LCM):
Multiples of 15:
Multiples of 20:
Common multiples of 15 and 20:
Least common multiple of 15 and 20:
Using mental math: Test _____________ of the _______ number: Can it be divided by the ______________
Example 1:
Example 2:
Find the LCM of 12 and 9
Q 1:
Find the LCM of 20 and 4
Q 2:
54
2.4c Least Common Multiple – LCM Using Prime Factorization
To find an LCM of two larger numbers:
1. Find the ______________________ of each
2. Use all the unique ___________________
3. Assign the _________________________ to each factor
Example 1:
Example 2:
Find the LCM of 24 and 36
Q 1:
Find the LCM of 54 and 90
Q 2:
55
2.4d Least Common Multiple – LCM with Variables
To find the LCM with variables:
1. Use all the unique ____________________
2. Assign the _________________________ to each variable
Example 1:
Find the LCM of 𝑎3 𝑏 2 𝑐 and 𝑎2 𝑏 7 𝑑2
Example 2:
Find the LCM of 6𝑥 2 𝑧 and 8𝑥 3 𝑦 2
Q 1:
Q 2:
You have completed the videos for 2.4 Least Common Multiple. On your own paper complete the
homework assignment.
56
2.5a Add and Subtract Fractions – With Common Denominator
2
1
Consider: 5 + 5
+ =
To add fractions that have the same denominator: _________ numerators and _____________ denominators
When adding fractions always check to ___________ at the ___________ of the problem
Example 1:
Example 2:
4 2
−
7 7
Q 1:
7
5
+
10 10
Q 2:
57
2.5b Add and Subtract Fractions – With Different Denominators
If the denominators don’t match we will find the ________________________
Multiply ______________________ by missing factors.
Then multiply the ___________________ by the ___________ factors
If you do not know the LCD you can always ______________ the two _______________
Example 1:
Example 2:
5 4
+
3 9
Q 1:
3 5
−
4 6
Q 2:
58
2.5c Add and Subtract Fractions – With Different Large Denominators
We may have to use _________________ to find the LCD.
To build up to the LCD we multiply by any _____________________ factors
Example 1:
Example 2:
7 11
+
24 36
Q 1:
5
7
−
54 90
Q 2:
You have completed the videos for 2.5 Add and Subtract Fractions. On your own paper complete the
homework assignment.
59
2.6a Order of Operations with Fractions – Exponents on Fractions
Exponents mean ______________________________
2 2
(3) =
Or we could put the exponent on the __________________ and ____________________
Example 1:
Example 2:
3
7 2
( )
4
3
( )
2
Q 1:
Q 2:
60
2.6b Order of Operations with Fractions – Order of Operations with Fractions
The order
1.
2.
3.
4.
You may need some _________________
Example 1:
Example 2:
2
8 2
9 7 9
( ) −
| − |
5
10 3 2
9 12
5
1
÷
+( ) ∙
10 5
2
30
61
Q 1:
Q 2:
You have completed the videos for 2.6 Order of Operations with Fractions. On your own paper
complete the homework assignment.
62
2.7a Mixed Numbers – Mixed Numbers and Conversions
Mixed number:
Change a mixed number to a fraction: ___________ the whole and ____________
and ___________ the __________________
Change a fraction to a mixed number: _______________, the remainder is the new _________________
Example 1:
Example 2:
9
73
Convert 5 11 to a fraction
Q 1:
Convert 12 to a mixed number
Q 2:
63
2.7b Mixed Numbers – Add and Subtract Mixed Numbers
To do math with mixed numbers it is easiest to _____________ to a _________________
When you have your answer, ________________________
Example 1:
Example 2:
2
3
5 +7
5
10
Q 1:
1
4
−2 + 6
3
9
Q 2:
64
2.7c Mixed Numbers – Multiply and Divide Mixed Numbers
To do math with mixed numbers it is easiest to _____________ to a _________________
When you have your answer, _______________________
Example 1:
Example 2:
4 4
2 ∙3
5 7
Q 1:
1
1
5 ÷2
3
6
Q 2:
You have completed the videos for 2.7 Mixed Numbers. On your own paper complete the
homework assignment.
Congratulations! You made it through the material for Unit 2: Fractions. It is time to prepare for
your exam. On a separate sheet of paper complete the practice test. Once you have completed the
practice test, ask your instructor to take the test. Good luck!
65
Unit 3:
Linear Equations
To work through the unit you should:
1.
2.
3.
4.
5.
6.
7.
Watch a video, as you watch, fill out the workbook (top and example sections).
Complete Q1 and Q2 in WAMAP, put your work in the bottom sections of the page.
Repeat #1 and #2 with each page until you reach
.
Complete the homework assignment on your own paper.
Repeat #1-#4 until you reach the end of the unit.
Complete the practice test on your own paper.
Take the unit exam.
66
3.1a One Step Equations – Addition Principle
A _______________ to an equation is the value for the ____________ that makes the equation ___________
We can _________ anything to __________________ of the equation
Addition Principle: To move a negative term we do the opposite and _________ it to ______________
Very Important to __________ your work!
Example 1:
Example 2:
𝑥−9=4
Q 1:
−3 = −5 + 𝑥
Q 2:
67
3.1b One Step Equations – Subtraction Principle
We can _________ anything to __________________ of the equation
Subtraction Principle: To move a positive term we do the opposite and __________ it from ______________
Very Important to __________ your work!
Example 1:
Example 2:
𝑥 + 8 = −4
Q 1:
3=7+𝑥
Q 2:
68
3.1c One Step Equations – Division Principle
We can _________ anything to __________________ of the equation
Division Principle: To undo multiplication of factors we do the opposite and _______________ it
from ______________
Very Important to __________ your work!
Example 1:
Example 2:
7𝑥 = 147
Q 1:
−8𝑥 = 72
Q 2:
69
3.1d One Step Equations – Multiplication Principle
We can ____________ anything to __________________ of the equation
Multiplication Principle: To clear division we do the opposite and _______________ it by ______________
Very Important to __________ your work!
Example 1:
Q 1:
𝑥
= −4
7
Example 2:
5=
𝑥
−2
Q 2:
You have completed the videos for 3.1 One Step Equations. On your own paper complete the
homework assignment.
70
3.2a Two Step Equations – Two Steps
Simplifying we use order of operations and we _____________________ before we ____________________
Solving we work __________________ and we _____________________ before we ____________________
Example 1:
Example 2:
5𝑥 − 7 = 8
Q 1:
−9 = −5 − 2𝑥
Q 2:
71
3.2b Two Step Equations – Negative Variables
If there is no number in front of a variable, we assume there is a _____ in front
This means −𝑥 is the same as _________
Example 1:
Example 2:
−𝑥 + 8 = 5
Q 1:
−4 = −6 − 𝑥
Q 2:
You have completed the videos for 3.2 Two Step Equations. On your own paper complete the
homework assignment.
72
3.3a General Linear Equations – Variable on Both Sides
When solving an equation we want the variable on _______________________________
If the variable is on both sides we will ________ the _____________ one by _______________________
Example 1:
Example 2:
5𝑥 + 7 = 9𝑥 − 2
Q 1:
−6𝑥 + 1 = 2𝑥 − 12
Q 2:
73
3.3b General Linear Equations – Combine Like Terms
Before we solve we must _______________ the __________ and ___________ sides
One way to do this is ________________________
Example 1:
Example 2:
5𝑥 − 3 − 2𝑥 = 7 + 8𝑥 − 1
Q 1:
4 + 𝑥 − 2 = −3𝑥 + 8 + 2𝑥
Q 2:
74
3.3c General Linear Equations – Distribute and Combine
Before we solve we must _______________ the __________ and ___________ sides
One way to do this is ________________________
Example 1:
2(3𝑥 − 1) = 4𝑥 + 6 − 𝑥
Q 1:
Example 2:
3(2𝑥 + 1) − 9𝑥 = 4(𝑥 + 6) − 20
Q 2:
You have completed the videos for 3.3 General Linear Equations. On your own paper complete the
homework assignment.
75
3.4a Equations with Decimals and Fractions – Decimals
When solving with decimals the pattern of solving is ____________________
A ______________ may be helpful to speed up calculations
Example 1:
3.2𝑥 + 7.11 = −19.77
Q 1:
Example 2:
2.1(𝑥 − 4.3) = 5.7𝑥 − 9.19 − 3.8𝑥
Q 2:
76
3.4b Equations with Decimals and Fractions – Clear Fractions with LCD
If the equation has fractions, we can clear the fractions by ____________________ each term by the _____
After multiplying we can _________________ to get an equation with no __________________
Example 1:
Example 2:
3
3
7
𝑥 + = −5 + 𝑥
8
4
2
5
1 7
𝑥− =
6
3 2
Q 1:
Q 2:
You have completed the videos for 3.4 Equations with Decimals and Fractions. On your own paper
complete the homework assignment.
Congratulations! You made it through the material for Unit 3: Linear Equations. It is time to prepare
for your exam. On a separate sheet of paper complete the practice test. Once you have completed
the practice test, ask your instructor to take the test. Good luck!
77
Unit 4:
Stats, Graphing,
Proportions and Percents
(You may use a calculator on this unit)
To work through the unit you should:
1.
2.
3.
4.
5.
6.
7.
Watch a video, as you watch, fill out the workbook (top and example sections).
Complete Q1 and Q2 in WAMAP, put your work in the bottom sections of the page.
Repeat #1 and #2 with each page until you reach
.
Complete the homework assignment on your own paper.
Repeat #1-#4 until you reach the end of the unit.
Complete the practice test on your own paper.
Take the unit exam.
78
4.1a Averages - Mean
Mean: The average if all items were the __________________ or spread out __________________
To calculate the mean:
Example 1:
Example 2:
Find the Mean:
5, 8, 6, 7, 9, 4, 8, 10
Q 1:
Find the Mean:
23, 26, 27, 21, 26, 22, 73, 24, 23
Q 2:
79
4.1b Averages – Missing Value
The mean is when all the items are the _________________ or spread out _____________________
If we know the mean and are missing a value, calculate the _________________ using the _______________
Example 1:
On three tests a student earns 83%, 71%, and 81%.
What must she earn on her fourth test to raise her
average of the four tests up to 80%?
Example 2:
Another student has a goal of 90% on his four tests.
On the first three tests he earned 92%, 75%, and
89%. Is it possible for him to reach his goal of 90%?
What score would he have to earn?
Q 1:
Q 2:
80
4.1c Averages – Weighted Mean
Weighted average: Values that occur _______________ have a larger _____________ on the average (mean)
To calculate the total we ________________ the _____________________ by the ___________________
Example 1:
In a survey, students were asked how many siblings
they had. The results are below. Calculate the average
number of siblings of the survey responders.
Siblings
0
1
2
3
4
Q 1:
Responses
8
38
21
15
2
Example 2:
Grade Point Average (GPA) is calculated as a
weighted average. The credits of a course are
considered the “frequency” of the course. In this
way, classes that are more credits have a larger
effect on grade than classes with fewer credits.
Calculate the GPA of the following report card:
Class
English
Math
History
PE
Q 2:
81
Credits
4
5
3
1
Grade
3.2
4.0
2.8
0.7
4.1d Averages - Median
Median: The average at which ___________ the data is _______________ and __________ is ____________
To calculate the median:
If two values are in the middle:
Example 1:
Example 2:
Find the Median:
5, 8, 6, 7, 9, 4, 8, 10
Q 1:
Find the Median:
23, 26, 27, 21, 26, 22, 73, 24, 23
Q 2:
82
4.1e Averages - Mode
Mode: the average or value that occurs _______________________
It is possible to have _______________ modes or ____ mode
Example 1:
Example 2:
Find the Mode:
5, 8, 6, 7, 9, 4, 8, 10
Q 1:
Find the Mode:
23, 26, 27, 21, 26, 22, 73, 24, 23
Q 2:
You have completed the videos for 4.1 Averages. On your own paper complete the homework
assignment.
83
4.2a Probability and Plotting Points – Basic Probability
Probability:
Basic Probability Fraction:
Example 1:
A bag contains 3 blue marbles, 2 red marbles and 1
green marble. If you were to draw one marble at
random, what is the probability of drawing...
Example 2:
If you roll a standard six-sided die, what is the
probability you roll…
1) A three?
1) A blue marble?
2) An even number?
2) A red marble?
3) A number smaller than three?
3) A black marble?
4) A seven?
Q 1:
Q 2:
84
4.2b Probability and Plotting Points – Compound Events
The probability of this OR that: we __________ the individual probabilities
The probability of this AND that: we _______________ the individual probabilities
Example 1:
A bag contains 3 blue marbles, 2 red marbles and 1
green marble. If you were to draw one marble at
random, what is the probability of drawing...
Example 2:
If you roll a standard six-sided die and then draw a
marble out of a bag with 7 red and 3 black marbles,
what is the probability you get…
1) A blue or green marble?
1) A three and a black?
2) A green or red marble?
2) An even and a red?
3) A blue or red or green marble?
Q 1:
Q 2:
85
4.2c Probability and Plotting Points – Give Coordinate
Coordinate Plane:
x-axis:
y-axis:
Origin:
Coordinate Point:
Example 1:
Give the coordinates of points A, B, and C
Example 2:
Give the coordinates of points E, F, and G
B
F
A
E
C
Q 1:
G
Q 2:
86
4.2d – Probability and Plotting Points – Plot Points
To plot a point start at the ______________________ and move ________________ then ____________
Negatives move the point _________________________________
Example 1:
Example 2:
Plot the points:
𝐴(2, −4), 𝐵(−3, −1), 𝐶(0,2), 𝐷(−3,4)
Q 1:
Plot the points:
𝐸(1,2), 𝐹(−1,0), 𝐺(0,0)
Q 2:
You have completed the videos for 4.2 Probability and Plotting Points. On your own paper complete
the homework assignment.
87
4.3a Rates and Unit Rates – Find Rates
Rate: Amount per ____________
To set up, the word ___________ is the ________________
Example 1:
Rafael made $22,512 last year. What is his rate of pay
per month?
Example 2:
Giovanni covered his 2,500 square foot yard with
700 ounces of fertilizer. What is the rate of coverage
the fertilizer can cover in ounces per square foot?
Q 1:
Q 2:
88
4.3b Rates and Unit Rates – Find Unit Rate
Unit Rate: Rate of _________________ per ________________________
Use unit rates to identify the ____________________
Example 1:
A 20 ounce bottle of soda sells for $1.99. What is the
unit price?
Example 2:
Lemon juice comes in a 24 oz bottle and a 32 oz
bottle. The 24 oz bottle sells for $1.98 and the 32 oz
bottle sells for $2.98. Which is the better deal and
what is the unit price?
Q 1:
Q 2:
You have completed the videos for 4.3 Rates and Unit Rates. On your own paper complete the
homework assignment.
89
4.4a Proportions and Applications – Solving Proportions
To solve a proportion we ________________ both sides by the ___________________
The quick method: Multiply by the __________________
Example 1:
Example 2:
7 6
=
𝑥 5
Q 1:
8 𝑥
=
5 3
Q 2:
90
4.4b Proportions and Applications – Proportion Applications
When solving applications we must first identify what we are ____________________
Clearly label the _____________________ and __________________________ of the proportion
Example 1:
A 65 inch tall man wants to determine how tall a large
tree is. He noticed at a certain time his shadow was
14 inches long. When he measured the shadow of the
tree he found it was 48 inches long. How tall is the
tree?
Example 2:
A manufacturer knows that out of every 300 parts
the company ships, on average 18 are defective. If
the company ships 5800 parts in a day, how many
will be defective?
Q 1:
Q 2:
You have completed the videos for 4.4 Proportions and Applications. On your own paper complete
the homework assignment.
91
4.5a Introduction to Percents – Convert Percents and Decimals
Percent:
To convert a decimal to a percent: Multiply by ______ or move the decimal ____________ to the _________
To convert a percent to a decimal: Divide by ______ or move the decimal ____________ to the _________
Example 1:
Example 2:
Convert 0.582 to a percent
Q 1:
Convert 145.6% to a decimal
Q 2:
92
4.5b Introduction to Percents – Convert Percents and Fractions
To convert a fraction to a percent: First _______________ then convert the ____________ to a ___________
To convert a percent to a fraction: Put the percent over __________ and _____________________
Example 1:
Example 2:
Convert
Q 1:
17
20
Convert 32% to a fraction
to a percent
Q 2:
You have completed the videos for 4.5 Introduction to Percents. On your own paper complete the
homework assignment.
93
4.6a Translate Percents and Applications – Translate and Solve
Key words to translate:

What

Is

Of

Percent
Example 1:
Example 2:
What is 70% of 40?
Q 1:
45% of what is 70?
Q 2:
94
4.6b Translate Percents and Applications - Discount
Discount Equation:
Example 1:
Example 2:
A computer that normally costs $549 is on sale at 22% A desk that normally sells for $224 is on sale for
off. What is the new price of the computer?
$188.16. What is the percent discount?
Q 1:
Q 2:
95
4.6c Translate Percents and Applications – Sales Tax
Sales Tax Equation:
Example 1:
What is the sales tax on a $499 television if the tax
rate is 8.5%? What would you pay for the TV?
Example 2:
Marc purchased a $390 table and paid $25.35 in
sales tax. What is the tax rate?
Q 1:
Q 2:
96
4.6d Translate Percents and Applications – Commission
Commission Equation:
Example 1:
A cell phone company pays a 14% commission to its
representatives. If an employee sells $23,000 worth
of product in a month, how much will she be paid?
Example 2:
A real estate agent sold $845,000 worth of
properties and earned $16,900 in his commission.
What is his commission rate?
Q 1:
Q 2:
97
4.6e Translate Percents and Applications – Simple Interest
Interest:
Simple Interest Equation:
Time must be in ______________
Example 1:
A bank account pays 2.1% simple interest on a certain
account. If you invest $3500 for 4 years, how much
will you earn in interest?
Example 2:
A bank gives a loan with 4.5% simple interest for 9
months on a $12,000 loan. How much is owed back
to the bank at the end of the loan?
Q 1:
Q 2:
You have completed the videos for 4.6 Translate Percents and Applications. On your own paper
complete the homework assignment.
98
4.7a Percents as Proportions and Applications – Translate and Solve
Percent Proportion:
Example 1:
Example 2:
What percent of 25 is 16?
Q 1:
14 is 60% of what?
Q 2:
99
4.7b Percents as Proportions and Applications – General Applications
“OF” represents ___________________
“IS” represents ____________________
Example 1:
Among male smokers, the lifetime risk of developing
lung cancer is 17.2%. According to the Washington
State Department of Health, in 2011 the state had
760,000 smokers. How many are at risk of developing
lung cancer in their lifetime?
Example 2:
In 2010, women made up 58% of Big Bend
Community College’s students. If there were 1688
women enrolled in 2010, how many students were
there total?
Q 1:
Q 2:
100
4.7c Percents as Proportions and Applications – Percent Increase/Decrease
Percent Increase/Decrease Proportions:
Example 1:
The price of a sofa was $299. During a weekend sale
the price was dropped to $179. What was the percent
decrease?
Example 2:
The population of a small town was 12,345 in 1990.
By 2000, the population was 31,416. What was the
percent increase?
Q 1:
Q 2:
You have completed the videos for 4.7 Percents as Proportions and Applications. On your own
paper complete the homework assignment.
Congratulations! You made it through the material for Unit 4: Stats, Graphing, Proportions, and
Percents. It is time to prepare for your exam. On a separate sheet of paper complete the practice
test. Once you have completed the practice test, ask your instructor to take the test. Good luck!
101
Unit 5:
Geometry and Intro to Polynomials
(You may use a calculator on this unit)
To work through the unit you should:
1.
2.
3.
4.
5.
6.
7.
Watch a video, as you watch, fill out the workbook (top and example sections).
Complete Q1 and Q2 in WAMAP, put your work in the bottom sections of the page.
Repeat #1 and #2 with each page until you reach
.
Complete the homework assignment on your own paper.
Repeat #1-#4 until you reach the end of the unit.
Complete the practice test on your own paper.
Take the unit exam.
102
Conversion Factors
LENGTH
English
Metric (meter)
12 in = 1 ft
3 ft = 1 yd
1 mi = 5280 ft
1000 mm = 1 m
100 cm = 1 m
10 dm = 1 m
1 dam = 10 m
1 hm = 100 m
1 km = 1000 m
TEMPERATURE
English to Metric
1 in = 2.54 cm
VOLUME
English
Metric (liter)
8 fl oz = 1 cup (c)
2 cups (c) = 1 pint (pt)
2 pints (pt) = 1 quart (qt)
4 quarts (qt) = 1 gallon (gal)
1000 mL = 1 L
100 cL = 1 L
10 dL = 1 L
1 daL = 10 L
1 hL = 100 L
1 kL = 1000 L
1 mL = 1 cc = 1 cm3
TIME
60 seconds (sec) = 1 minute (min)
60 minutes (min) = 1 hour (hr)
24 hours (hr) = 1 day
52 weeks = 1 year
365 days = 1 year
English to Metric
1 gallon (gal) = 3.79 liter (L)
1 in3 = 16.39 mL
WEIGHT (MASS)
English
Metric (gram)
16 oz = 1 pound (lb)
2,000 lb = 1 Ton (T)
1000 mg = 1 g
100 cg = 1 g
10 dg = 1 g
1 dag = 10 g
1 hg = 100 g
1 kg = 1000 g
INTEREST
.
Simple:
Continuous:
A = Pert
Compound:
Annual: n=1
Semiannual: n=2
English to Metric
Quarterly: n=4
2.20 lb = 1 kg
103
.
5.1a Convert Units – One Step Conversions
57 𝑖𝑛
Consider: (
1
1 𝑓𝑡
) (12𝑖𝑛) =
Divide out units by placing them in the __________________ part of the fraction
Conversion factor: Same _________ in numerator and denominator, but different _________
Dimensional analysis: Multiply by a _____________________ to convert units
Example 1:
Example 2:
17.2 in = _____ cm
Q 1:
88 lbs = _____kg
Q 2:
104
5.1b Convert Units – Multi-Step Conversions
If we do not have the correct conversion factor we can convert using _______________ conversion factors
Example 1:
Example 2:
365 g = _____ lbs
Q 1:
5 gal = _____ cups
Q 2:
105
5.1c Convert Units – Dual Unit Conversions
Dual Unit:
“Per” is the _________________________
With dual units we convert ___________________________
Example 1:
Example 2:
45 oz per min = _____ lbs per hr
Q 1:
35 mi per hr = _____ ft per sec
Q 2:
You have completed the videos for 5.1 Convert Units. On your own paper complete the homework
assignment.
106
Geometric Formulas
Name
Rectangle
Diagram
Area
w
A = lw
l
h
Parallelogram
A = bh
b
Triangle
h
b
a
h
Trapezoid
P = 2l + 2w
A=
bh
A=
h(a + b)
b
d
Circle
Name
r
A = πr2
Diagram
C = πd = 2πr
Volume
Rectangular
Solid
h
l
w
V = lwh
r
Right Circular
Cylinder
V = πr2h
h
Right Circular
Cone
h
V=
πr2h
V=
πr3
r
Sphere
r
Right Triangle
Pythagorean Theorem:
a2
+
b2
=
c2
107
c
a
b
5.2a Area, Volume, and Temperature – Area of Rectangle and Parallelogram
Area of a rectangle:
Area of a parallelogram:
Important: height must meet the base at a perfect ___________ angle
Example 1:
Example 2:
Find the area:
Find the area:
8 in
7 cm
21 cm
15 in
Q 1:
Q 2:
108
9 cm
5.2b Area, Volume, and Temperature – Area of Triangle
Area of a triangle:
Important: height must meet the base at a perfect ___________ angle
Example 1:
Example 2:
Find the area:
Find the area:
3m
5m
5 ft
4.3 m
8 ft
Q 1:
6m
Q 2:
109
5.2c Area, Volume, and Temperature – Area of Trapezoid
Area of a trapezoid:
The bases (a and b) must be __________________
The _____________ connects the two _________________
Important: height must meet the base at a perfect ___________ angle
Example 1:
Example 2:
Find the area:
Find the area:
12 cm
5 in
5 cm
4 cm
9 in
5 cm
2 in
6 cm
4 in
Q 1:
Q 2:
110
5.2d Area, Volume, and Temperature – Area of Circle
Diameter:
Radius:
𝜋=
Area of a Circle:
Example 1:
Example 2:
Find the area:
Find the area:
12 cm
4 ft
Q 1:
Q 2:
111
5.2e Area, Volume, and Temperature – Composition of Shapes
If we do not have a formula for a shape we must ___________________________
Shapes attached to each other are _________________
Shapes cut out are _____________________
Example 1:
Example 2:
Find the area:
Find the area:
2 mm
4 in
6 in
9 in
Q 1:
Q 2:
112
5.2f Area, Volume, and Temperature – Volume of a Rectangular Solid
Volume of rectangular solid:
Example 1:
Example 2:
On Southwest Airlines, the maximum size of a carryon bag is a length of 24 inches, a width of 10 inches
and a height of 16 inches. How much is packed in
this maximum sized bag?
Find the volume:
14
3
3
24
7
Q 1:
2
ft
ft
ft
Q 2:
113
5.2g Area, Volume, and Temperature – Volume of a Cylinder
Volume of a cylinder:
Example 1:
Example 2:
A can of soda is about 5 inches tall with a diameter
of 2.5 inches. How much soda can fit into the can?
Find the volume:
6 ft
8 ft
Q 1:
Q 2:
114
5.2h Area, Volume, and Temperature – Volume of a Cone
Volume of a cone:
Example 1:
Example 2:
How much ice cream can fit inside a cone that is 5
cm tall and 3 cm wide?
Find the volume:
24 ft
Q 1:
2 ft
Q 2:
115
5.2i Area, Volume, and Temperature – Volume of a Sphere
Volume of a sphere:
Example 1:
Find the volume:
Example 2:
Find the volume of the “planet” Pluto if the radius is
approximately 1195 km.
9 ft
Q 1:
Q 2:
116
5.2j Area, Volume, and Temperature – Composition of Solids
If we do not have a formula for a shape we must ___________________________
Example 1:
Find the volume
4 ft
10 ft
Q 1:
117
5.2k Area, Volume, and Temperature – Convert Temperature
Formulas to convert temperature:
𝐶=
𝐹=
Example 1:
98.6ᵒ F is body temperature. Find body temperature
in degrees Celsius.
Example 2:
100ᵒ C is the boiling point of water. Find at what
temperature water boils in degrees Fahrenheit.
Q 1:
Q 2:
You have completed the videos for 5.2 Area, Volume, and Temperature. On your own paper
complete the homework assignment.
118
5.3a Pythagorean Theorem – Square Roots
Square root asks the question what number ______________ is the inside number?
√25 =
On calculator use the _____ button (may have to use the ______ or ______ button first)
Example 1:
Example 2:
√225
Q 1:
√456
Q 2:
119
5.3b Pythagorean Theorem – Find Missing Side
Name the sides of the right triangle:
Pythagorean Theorem:
C is always the ______________________
When finding a leg with the Pythagorean theorem, first _________________ then _______________
Example 1:
Example 2:
Find the missing side:
Find the missing side:
4 cm
8 cm
7 yds
2 yds
Q 1:
Q 2:
120
5.3c Pythagorean Theorem – Applications
With applications it is always helpful to __________________________________
Example 1:
The base of a ladder is four feet from a building. The
top of the ladder is eight feet up the building. How
long is the ladder?
Example 2:
A young boy is flying a kite. He let out 21 meters of
string until the kite was flying over the head of his
sister who was 9 meters away. How high is the kite?
Q 1:
Q 2:
You have completed the videos for 5.3 Pythagorean Theorem. On your own paper complete the
homework assignment.
121
5.4a Introduction to Polynomials – Add Polynomials
Term:
Monomial:
Binomial:
Trinomial:
Polynomial:
Adding Polynomials: _________________ the _________________________
Example 1:
(4𝑥 3 − 2𝑥 2 + 𝑥) + (−3𝑥 2 − 5𝑥 + 7)
Example 2:
(3𝑎𝑏 3 − 2𝑎2 𝑏 + 𝑎𝑏) + (4𝑎2 𝑏 − 2𝑎𝑏 3 + 4𝑎𝑏)
Q 1:
Q 2:
122
5.4b Introduction to Polynomials – Subtract Polynomials
First ________________ the _________________.
Then ______________________________
Example 1:
(3𝑥 2 − 7𝑥 + 8) − (2𝑥 2 + 9𝑥 − 4)
Example 2:
(2𝑥 − 8𝑦 + 6) − (−3𝑦 − 7 + 5𝑥)
Q 1:
Q 2:
123
5.4c Introduction to Polynomials – Multiply Monomials
Consider 𝑎3 ∙ 𝑎2 =
When multiplying variables we __________ the exponents
If there is no exponent, then we assume the exponent is a ___
Example 1:
Example 2:
(4𝑥 3 𝑦 2 𝑧)(2𝑥 7 𝑦𝑧 4 )
Q 1:
−7𝑎3 𝑏 2 𝑐 4 ∙ 2𝑎𝑏 8 𝑐 4
Q 2:
124
5.4d Introduction to Polynomials – Multiply a Monomial by a Polynomial
Consider: 5(3𝑥 − 6) =
When multiplying a monomial by a polynomial we ___________________
Example 1:
Example 2:
5𝑥 3 (3𝑥 2 − 4𝑥 + 2)
Q 1:
−2𝑎3 𝑏(5𝑎𝑏 4 − 6𝑎2 𝑏 7 + 2𝑎4 )
Q 2:
You have completed the videos for 5.4 Introduction to Polynomials. On your own paper complete
the homework assignment.
125
5.5a Scientific Notation – Convert Scientific Notation to Standard Notation
Scientific Notation: 𝑎 × 10𝑏
𝑎 is greater than or equal to ____ but less than _____. This means the decimal is always after the
_____________ digit
𝑏 tells us how many times to ___________ the decimal
If 𝑏 is negative, then standard notation is ____________. If 𝑏 is positive, then standard notation is _________
Example 1:
Example 2:
Convert to standard notation:
4.21 × 105
Q 1:
Convert to standard notation:
6.2 × 10−3
Q 2:
126
5.5b Scientific Notation – Convert Standard Notation to Scientific Notation
To convert to scientific notation _____________ the number of times the _______________ must move
If standard notation is small then the exponent is ____________ if it is big then the exponent is ___________
When converting we will __________ the extra ________________
Example 1:
Example 2:
Convert to scientific notation:
48,100,000,000
Q 1:
Convert to scientific notation
0.0000235
Q 2:
127
You have completed the videos for 5.5 Scientific Notation. On your own paper complete the
homework assignment.
Unit 6:
Proficiency Exam #1
To work through this unit you should:
1. Complete the practice tests on your own paper.
2. Take the (three part) unit exam.
128
Unit 7:
Linear Equations and Applications
To work through the unit you should:
1.
2.
3.
4.
5.
6.
7.
Watch a video, as you watch, fill out the workbook (top and example sections).
Complete Q1 and Q2 in WAMAP, put your work in the bottom sections of the page.
Repeat #1 and #2 with each page until you reach
.
Complete the homework assignment on your own paper.
Repeat #1-#4 until you reach the end of the unit.
Complete the practice test on your own paper.
Take the unit exam.
129
7.1a Order of Operations – The Order
The Order:
1.
2.
3.
4.
To remember:
Example 1:
Example 2:
2
30 ÷ 5(−2) + (4 − 7)2
5 − 3(2 + 4 )
Q 1:
Q 2:
130
7.1b Order of Operations – Lots of Parentheses
Different types of parenthesis:
Always do _____________________________ first!
Example 1:
Example 2:
2
7{2 + 2[20 ÷ (4 + 6)]}
(4 + 2) − [5 ÷ (2 + 3)]
Q 1:
Q 2:
131
7.1c Order of Operations – Fractions
When simplifying fractions, always simplify _________________ and _________________ first
Only reduce at after the rest has been ______________________
Example 1:
Example 2:
−42 − (4 + 2 ∙ 3)
5 + 3(5 − 4)
(4 + 5)(2 − 9)
23 − (22 + 3)
Q 1:
Q 2:
132
7.1d Order of Operations – Absolute Value
Absolute values works just like ______________ but makes the number inside __________
after it has been _______________
Example 1:
Example 2:
4
2
2 − 4|32 + (52 − 62 )|
−3|2 − (5 + 4) |
Q 1:
Q 2:
You have completed the videos for 7.1 Order of Operations. On your own paper complete the
homework assignment.
133
7.2a Evaluate and Simplify Algebraic Expressions – Substitute a Value
Replace the ________________ with what it _____________________
Whenever we make a substitution or __________________ put it in _______________________
Example 1:
Example 2:
2
Evaluate 4𝑥 − 3𝑥 + 2
When 𝑥 = −3
Q 1:
Evaluate 4𝑏(2𝑥 + 3𝑦)
When 𝑏 = −2, 𝑥 = 5, 𝑦 = −7
Q 2:
134
7.2b Evaluate and Simplify Algebraic Expressions – Combine Like Terms
Terms are _________ and _____________ that are __________________ together
Like terms are terms that have matching ______________ and _________________________
Combine like terms: _____________ the coefficients from the _______________________.
Example 1:
4𝑥 3 − 2𝑥 2 + 5𝑥 3 + 2𝑥 − 4𝑥 2 − 6𝑥
Example 2:
Q 1:
Q 2:
4𝑦 − 2𝑥 + 5 − 6𝑦 + 7𝑦 − 9
135
7.2c Evaluate and Simplify Algebraic Expressions – Distributive Property
Distributive Property: 𝑎(𝑏 + 𝑐) =
Example 1:
Example 2:
4(7𝑥 2 − 6𝑥 + 1)
−2(5𝑥 − 4𝑦 + 3)
Q 1:
Q 2:
136
7.2d Evaluate and Simplify Algebraic Expressions – Distribute and Combine Like Terms
Order of operations states we ______________________ before we __________________
Therefore we will __________________ first and then ___________________________ second
Example 1:
Example 2:
4(3𝑥 − 7) − 7(2𝑥 − 1)
Q 1:
2(7𝑥 − 3) − (8𝑥 + 9)
Q 2:
You have completed the videos for 7.2 Evaluate and Simplify Algebraic Expressions. On your own
paper complete the homework assignment.
137
7.3a Solve Linear Equations – Variable on Both Sides
Move the variable to one side by ____________________
Solve remaining two step equation by _______________ first and ___________________ second.
Example 1:
Example 2:
−3𝑥 + 4 = 16 − 8𝑥
Q 1:
2𝑥 − 7 = 8𝑥 − 9
Q 2:
138
7.3b Solve Linear Equations – Simplify First
The first step of solving is to __________________ each side ___________________
We can simplify by ________________ and _________________________________
Example 1:
3(2𝑥 − 6) + 8 = 17
Q 1:
Example 2:
12𝑥 − 5(3𝑥 − 1) = 4 + 3(2𝑥 + 1)
Q 2:
139
7.3c Solve Linear Equations - Fractions
Clear fractions by ______________ by the ______
Be sure to multiply _________ term on _______ sides
Example 1:
Example 2:
3
7
7
𝑥−
= −4 + 𝑥
5
10
15
3
1 5
𝑥− =
4
2 6
Q 1:
Q 2:
140
7.3d Solve Linear Equations – Special Cases
Sometimes the variable ____________________!
This means there is either __________________________ or _______________________________
Example 1:
Example 2:
3𝑥 − 9 = 3(𝑥 − 3)
2𝑥 + 5 = 2𝑥 − 1
Example 3:
Example 4:
6𝑥 + 2 = 3(2𝑥 + 1)
Q 1:
4𝑥 + 1 = 2(2𝑥 + 3) − 5
Q 2:
You have completed the videos for 7.3 Solve Linear Equations. On your own paper complete the
homework assignment.
141
7.4a Formulas – Two Step Formulas
Solving formulas: Treat other variables like _______________
Final answer is an ________________
Example: 3𝑥 = 15
and
𝑤𝑥 = 𝑦
Example 1:
Example 2:
Solve 𝑤𝑥 + 𝑏 = 𝑦 for 𝑥
Q 1:
Solve 𝑎𝑏 + 5𝑦 = 𝑤𝑥 + 𝑦 for 𝑏
Q 2:
142
7.4b Formulas – Multi-Step Formulas
Strategy:
IMPORTANT: Terms ________________ reduce
Example 1:
Solve 𝑎(3𝑥 + 𝑏) = 𝑏𝑦 for 𝑥
Q 1:
Example 2:
Solve 3(𝑎 + 2𝑏) + 5𝑏 = −2𝑎 + 𝑏 for 𝑎
Q 2:
143
7.4c Formulas – Fractions and Formulas
Clear fractions by _____________________________
Example 1:
Example 2:
5
1
𝑏
Q 1:
1
Solve 𝐴 = 2 ℎ𝑏1 + 2 ℎ𝑏2 for 𝑏1
Solve 𝑥 + 4𝑎 = 𝑥 for 𝑥
Q 2:
144
7.4d Formulas – Factoring the Variable
Distributive property in reverse (Factor): 𝑎𝑏 + 𝑎𝑐 =
Put all terms with the variable one ________________ and the other terms on the __________________
Factor out the _________________ and then ______________ to isolate
Example 1:
Example 2:
Solve 𝐴 = 𝜋𝑟 2 + 𝜋𝑟𝑙 for 𝜋
Solve 𝑎𝑥 + 𝑏 = 𝑐𝑥 + 𝑑 for 𝑥
Q 1:
Q 2:
You have completed the videos for 7.4 Formulas. On your own paper complete the homework
assignment.
145
7.5a Absolute Value Equations – Two Solutions
|𝑥| = 5 so the 𝑥 could be ____ or ____
What is inside the absolute value can be ___________________ or ____________________
This means we have _____________________________
Example 1:
Example 2:
|2𝑥 − 5| = 7
Q 1:
|7 − 5𝑥| = 17
Q 2:
146
7.5b Absolute Value Equations – Isolate the Absolute Value
Before we look at our two equations, we must first ___________________________
Never _______________ through absolute value!
Never ___________ a term ________ an absolute value and a term ____________ an absolute value!
Example 1:
Example 2:
5 + 2|3𝑥 − 4| = 11
Q 1:
−3 − 7|2 − 4𝑥| = −31
Q 2:
147
7.5c Absolute Value Equations – Dual Absolute Values
With two absolutes, we need _______________________
The first equation is ___________________________
The second equation is _______________________________
Example 1:
Example 2:
|2𝑥 − 6| = |4𝑥 + 8|
Q 1:
|3𝑥 − 5| = |7𝑥 − 2|
Q 2:
You have completed the videos for 7.5 Absolute Value Equations. On your own paper complete the
homework assignment.
148
7.6a Word Problems - Number
Translate:

Is/Were/Was/Will Be:

More than:

Subtracted from/Less than:
Example 1:
Five less than three times a number is nineteen. What
is the number?
Example 2:
Seven more than twice a number is six less than
three times the same number. What is the number?
Q 1:
Q 2:
149
7.6b Word Problems – Consecutive Integers
Consecutive Numbers:
First:
Second:
Third:
Example 1:
Find three consecutive numbers whose sum is 543.
Example 2:
Find four consecutive integers whose sum is -222.
Q 1:
Q 2:
150
7.6c Word Problems – Consecutive Even/Odd Integers
Consecutive Even:
Consecutive Odd:
First:
First:
Second:
Second:
Third:
Third:
Example 1:
Find three consecutive even integers whose sum is 84
Example 2:
Find four consecutive odd integers whose sum is 152
Q 1:
Q 2:
151
7.6d Word Problems - Triangles
The angles of a triangle add to ___________
Example 1:
Two angles of a triangle are the same measure. The
third angle is 30 degrees less than the first. Find the
three angles.
Example 2:
The second angle of a triangle measures twice the
first. The third angle is 30 degrees more than the
second. Find the three angles.
Q 1:
Q 2:
152
7.6e Word Problems - Perimeter
Formula for perimeter of a rectangle:
Width is the ____________ side
Example 1:
A rectangle is three times as long as it is wide. If the
perimeter is 112 cm, what is the length?
Example 2:
The width of a rectangle is 6 cm less than the length.
If the perimeter is 52 cm, what is the width?
Q 1:
Q 2:
You have completed the videos for 7.6 Word Problems. On your own paper complete the homework
assignment.
153
7.7a Age Problems – Unknown Now
Table:
Equation is always for the _______________________
Example 1:
Alexis is five years younger than Brian. In seven years
the sum of their ages will be 49 years. How old is each
now?
Example 2:
Maria is ten years older than Sonia. Eight years ago
Maria was three times Sonia’s age. How old is each
now?
Q 1:
Q 2:
154
7.7b Age Problems – Given the Sum Now
Consider: Sum of 8…..
When we have the sum now, for the first box we use ____ and the second we use _________________
Example 1:
The sum of the ages of a man and his son is 82 years.
How old is each if 11 years ago, the man was twice his
son’s age?
Example 2:
The sum of the ages of a woman and her daughter is
38 years. How old is each if the woman will be triple
her daughter’s age in 9 years?
Q 1:
Q 2:
155
7.7c Age Problems – Unknown Time
If we don’t know the time:
Example 1:
A man is 23 years old. His sister is 11 years old. How
many years ago was the man triple his sister’s age?
Example 2:
A woman is 11 years old. Her cousin is 32 years old.
How many years until her cousin is double her age?
Q 1:
Q 2:
You have completed the videos for 7.7 Age Problems. On your own paper complete the homework
assignment.
156
7.8a Distance, Rate, Time Problems – Opposite Directions
The distance equation:
Opposite directions:
OR
Example 1:
Brian and Jennifer both leave the convention at the
same time, traveling in opposite directions. Brian
drove 35 mph and Jennifer drove 50 mph. After how
much time were they 340 miles apart?
Example 2:
Maria and Tristan are 126 miles apart biking towards
each other. If Maria bikes 6 mph faster than Tristan
and they meet after 3 hours, how fast did each ride?
Q 1:
Q 2:
157
7.8b Distance, Rate, Time Problems – Catch Up
_________ the head start to the ________________ of the person with the head start
Catch up:
Example 1:
Raquel left the party traveling 5 mph. Four hours later
Nick left to catch up with her, traveling 7 mph. How
long will it take him to catch up with her?
Example 2:
Trey left on a trip traveling 20 mph. Julian left 2
hours later, traveling in the same direction at 30
mph. After how many hours does Julian pass Trey?
Q 1:
Q 2:
158
7.8c Distance, Rate, Time Problems – Total Time
Consider: Total time of 8…..
When we have the total time, we use ____ for the first time and _________________ for the second
Example 1:
Lupe rode into the forest at 10 mph, turned around
and returned by the same route traveling 15 mph. If
her trip took 5 hours, how long did she travel at each
rate?
Example 2:
Ian went on a 230 mile trip. He started driving 45
mph. However, due to construction on the second
leg of the trip, he had to slow down to 25 mph. If the
trip took 6 hours, how long did he drive at each
speed?
Q 1:
Q 2:
You have completed the videos for 7.8 Distance, Rate, Time Problems. On your own paper complete
the homework assignment.
159
7.9a Inequalities - Graphing
Inequalities:

Less than:

Less than or equal to:

Greater than:

Greater than or equal to:
Graphing on number line: Use
Example 1:
for less/greater than and use
Example 2:
Graph 𝑥 ≥ −3
Q 1:
when its “or equal to”
Give the inequality
Q 2:
160
7.9b Inequalities – Interval Notation
Interval notation: (
Use
,
)
for less/greater than and use
when its “or equal to”
∞ and −∞ always use a
Example 1:
Example 2:
Graph the interval (−∞, −1)
Give interval notation
Q 1:
Q 2:
161
7.9c Inequalities – Solving
Solving inequalities is very similar to solving _____________________________ (with one exception…)
Three steps with inequalities: ______________, then __________________, then _____________________
Example 1:
Example 2:
7 + 5𝑥 ≤ 17
Q 1:
3(𝑥 + 8) + 2 > 5𝑥 − 20
Q 2:
162
7.9d Inequalities – Multiply or Divide by a Negative
What happens to 5 > −2 when we multiply both sides by −3?
(−3)5 ? ? −2(−3)
When __________________ or ______________________ by a ________________ you must
________________________________________
Three steps with inequalities: ______________, then __________________, then _____________________
Example 1:
Example 2:
7 − 3𝑥 ≤ 16
Q 1:
4 < −2𝑥 + 16
Q 2:
163
7.9e Inequalities - Tripartite
Tripartite inequalities:
When solving __________________________
When graphing _________________________
Three steps with inequalities: ______________, then __________________, then _____________________
Example 1:
Example 2:
2 ≤ 5𝑥 + 7 < 22
Q 1:
5 < 5 − 4𝑥 ≤ 13
Q 2:
You have completed the videos for 7.9 Inequalities. On your own paper complete the homework
assignment.
Congratulations! You made it through the material for Unit 7: Linear Equations and Applications. It
is time to prepare for your exam. On a separate sheet of paper complete the practice test. Once you
have completed the practice test, ask your instructor to take the test. Good luck!
164
Unit 8:
Graphing Linear Equations
To work through the unit you should:
1.
2.
3.
4.
5.
6.
7.
Watch a video, as you watch, fill out the workbook (top and example sections).
Complete Q1 and Q2 in WAMAP, put your work in the bottom sections of the page.
Repeat #1 and #2 with each page until you reach
.
Complete the homework assignment on your own paper.
Repeat #1-#4 until you reach the end of the unit.
Complete the practice test on your own paper.
Take the unit exam.
165
8.1a Slope – Graphing Points and Lines
The Coordinate Plane:
Give ________________________ to a point going _______________ then _________________ as ______
If we have an equation we can pick values for ______ and find values for ___
Example 1:
Example 2:
Graph the line 𝑦 = 2𝑥 − 1
Graph the points
(−2,3), (4, −1), (−2, −4), (0,3), (−1,0) and (3,4)
Q 1:
Q 2:
166
8.1b Slope – Slope from Graph
Slope:
Negative Slope:
Positive Slope:
Example 1:
Big Slope:
Example 2:
Find the Slope
Q 1:
Small Slope:
Find the Slope
Q 2:
167
8.1c Slope – Slope from Points
Slope Equation:
Example 1:
Find the slope between (7,2) and (11,4)
Example 2:
Find the slope between (−2, −5) and (−17,4)
Q 1:
Q 2:
You have completed the videos for 8.1 Slope. On your own paper complete the homework
assignment.
168
8.2a Equations of Lines – Slope-Intercept Equation
Slope Intercept Equation:
Example 1:
Give the equation of the line with
3
a slope of − 4 and a y-intercept of 2
Example 2:
Give the equation of the graph:
Q 1:
Q 2:
169
8.2b Equations of Lines – Equation Through a Point
To find the y-intercept we use _________________ and solve for _____________
Example 1:
Give the equation of the line that passes
through (6, −2) and has a slope of 4.
Example 2:
Give the equation of the line that passes
2
through (−3,5) and has a slope of − 3
Q 1:
Q 2:
170
8.2c Equations of Lines – Put in Slope-Intercept Form
We may have to put the equation in _________________________
To do this we ________________________________
Example 1:
Example 2:
Give the slope and y-intercept
5𝑥 + 8𝑦 = 16
Q 1:
Give the slope and y-intercept
−3𝑥 + 2𝑦 = 8
Q 2:
171
8.2d Equations of Lines – Graph a Linear Equation
We can graph an equation by identifying the ____________________ and _________________________
Start at the __________________ and use the __________________________ for changing to the next point
Remember slope is ________________ over _______________
Example 1:
Example 2:
3
Graph 3𝑥 − 2𝑦 = 2
Graph 𝑦 = − 4 𝑥 + 2
Q 1:
Q 2:
172
8.2e Equations of Lines – Given Two Points
To find the equation of a line you must have the ________________________
Recall the slope formula:
To find the y-intercept we use _________________ and solve for _____________
Example 1:
Example 2:
Find the equation of the line
through (−3, −5) and (2,5)
Q 1:
Find the equation of the line
through (1, −4) and (3,5)
Q 2:
You have completed the videos for 8.2 Equations of Lines. On your own paper complete the
homework assignment.
173
8.3a Parallel and Perpendicular Lines - Slopes
Parallel Lines:
Perpendicular Lines:
Slope:
Slope:
Neither:
Example 1:
One line goes through (5,2) and (7,5). Another line
goes through (−2, −6) and (0, −3). Are the lines
parallel, perpendicular, or neither?
Example 2:
One line goes through (−4,1) and (−1,3). Another
line goes through (2, −1) and (6, −7). Are the lines
parallel, perpendicular, or neither?
Example 3:
One line goes through (3,7) and (−6, −8). Another
line goes through (5,2) and (−5, −4). Are the lines
parallel, perpendicular, or neither?
Q 1:
Q 2:
Q 3:
174
8.3b Parallel and Perpendicular Lines – Parallel Equations
Parallel lines have the _________ slope
Once we know the slope and a point we can use the formula:
Example 1:
Find the equation of the line parallel to the line 𝑦 =
3
− 4 𝑥 + 2 that goes through the point (−8,1)
Example 2:
Find the equation of the line parallel to the line 2𝑥 −
5𝑦 = 3 that goes through the point (5,3)
Q 1:
Q 2:
175
8.3c Parallel and Perpendicular Lines - Perpendicular Equations
Perpendicular lines have __________________________ slopes
Once we know the slope and a point we can use the formula:
Example 1:
Find the equation of the line perpendicular to the line
𝑦 = 5𝑥 + 1 that goes through the point (−5,2)
Example 2:
Find the equation of the line perpendicular to the
line 3𝑥 + 2𝑦 = 5 that goes through the point
(−3, −4)
Q 1:
Q 2:
You have completed the videos for 8.2 Parallel and Perpendicular Lines. On your own paper
complete the homework assignment.
Congratulations! You made it through the material for Unit 8: Graphing Linear Equations. It is time
to prepare for your exam. On a separate sheet of paper complete the practice test. Once you have
completed the practice test, ask your instructor to take the test. Good luck!
176
Unit 9:
Polynomials
To work through the unit you should:
1.
2.
3.
4.
5.
6.
7.
Watch a video, as you watch, fill out the workbook (top and example sections).
Complete Q1 and Q2 in WAMAP, put your work in the bottom sections of the page.
Repeat #1 and #2 with each page until you reach
.
Complete the homework assignment on your own paper.
Repeat #1-#4 until you reach the end of the unit.
Complete the practice test on your own paper.
Take the unit exam.
177
9.1a Exponents – Product Rule
𝑎3 ∙ 𝑎2 =
Product Rule: 𝑎𝑚 ∙ 𝑎𝑛 =
Example 1:
Example 2:
3
2
(5𝑎3 𝑏 7 )(2𝑎9 𝑏 2 𝑐 4 )
(2𝑥 )(4𝑥 )(−3𝑥)
Q 1:
Q 2:
178
9.1b Exponents – Quotient Rule
𝑎5
𝑎3
=
Quotient Rule:
𝑎𝑚
𝑎𝑛
=
Example 1:
Example 2:
8𝑚7 𝑛4
−6𝑚5 𝑛
7 2
𝑎 𝑏
𝑎3 𝑏
Q 1:
Q 2:
179
9.1c Exponents – Power Rules
(𝑎𝑏)3 =
Power of a Product: (𝑎𝑏)𝑚 =
𝑎 3
(𝑏 ) =
𝑎 𝑚
=
Power of a Quotient: ( )
𝑏
(𝑎2 )3 =
Power of a Power: (𝑎𝑚 )𝑛 =
Example 1:
Example 2:
(5𝑎4
Q 1:
𝑏)
2
3
−5𝑚3
(
)
9𝑛4
Q 2:
180
9.1d Exponents – Zero Exponent
𝑎3
𝑎3
=
Zero Power Rule: 𝑎0 =
Example 1:
Example 2:
(5𝑥 3
Q 1:
𝑦𝑧
(3𝑥 2 𝑦 0 )(5𝑥 0 𝑦 4 )(𝑥 2 𝑦 3 )
5 )0
Q 2:
181
9.1e Exponents – Negative Exponents
𝑎3
𝑎5
=
Negative Exponent Rules: 𝑎−𝑚 =
Example 1:
1
𝑎−𝑚
=
=
Example 2:
2
5𝑎−4
Q 1:
𝑎 −𝑚
(𝑏 )
7𝑥 −5
3−1 𝑦𝑧 −4
Q 2:
182
9.1f Exponents - Properties
𝑎𝑚 𝑎𝑛 =
𝑎𝑚
=
𝑎𝑛
(𝑎𝑏)𝑚 =
𝑎 𝑚
( ) =
𝑏
(𝑎𝑚 )𝑛 =
𝑎0 =
𝑎−𝑚 =
1
𝑎−𝑚
=
𝑎 −𝑚
( )
=
𝑏
To Simplify:
Example 1:
Example 2:
(2𝑥 2 𝑦 −3 )−4 (𝑥 4 𝑦 −6 )−2
(𝑥 −6 𝑦 4 )2
(4𝑥 −5 𝑦 2 𝑧)2 (2𝑥 4 𝑦 −2 𝑧 3 )4
Q 1:
Q 2:
You have completed the videos for 9.1 Exponents. On your own paper complete the homework
assignment.
183
9.2a Scientific Notation – Convert Scientific and Standard Notation
𝑎 × 10𝑏
𝑎 is
𝑏 is
𝑏 positive
𝑏 negative
Example 1:
Example 2:
Convert to Standard Notation
5.23 × 105
Example 3:
Convert to Standard Notation
4.25 × 10−4
Example 4:
Convert to Scientific Notation
81,500,000
Convert to Scientific Notation
0.0000245
Q 1:
Q 2:
Q 3:
Q 4:
184
9.2b Scientific Notation – Almost Scientific Notation
Put the number in front in ___________________________
Then use ______________________________ on the 10’s
Example 1:
Example 2:
−8
0.0032 × 105
523.6 × 10
Q 1:
Q 2:
185
9.2c Scientific Notation – Multiply or Divide
Multiply/Divide the _____________________________
Then use ______________________________ on the 10’s
Example 1:
Example 2:
5
5.32 × 104
1.9 × 10−3
−2
(3.4 × 10 )(2.7 × 10 )
Q 1:
Q 2:
186
9.2d Scientific Notation – Multiply or Divide where Answer is not in Scientific Notation
If our final answer is not in scientific notation we must _______________________
Example 1:
Example 2:
−6
−3
2.352 × 10−6
8.4 × 10−2
(6.7 × 10 )(5.2 × 10 )
Q 1:
Q 2:
187
9.2e Scientific Notation – Multiply and Divide
Multiply/Divide the _____________________________
Then use ______________________________ on the 10’s
Example 1:
Example 2:
4 )(8.1
(4.2 × 10
× 10
1.4 × 105
Q 1:
−6 )
2.01 × 10−5
(1.5 × 10−3 )(3.2 × 10−4 )
Q 2:
You have completed the videos for 9.2 Scientific Notation. On your own paper complete the
homework assignment.
188
9.3a Add, Subtract, Multiply Polynomials - Evaluate
Term:
Monomial:
Binomial:
Trinomial:
Polynomial:
Evaluate:
Example 1:
Example 2:
5𝑥 2 − 2𝑥 + 6 when 𝑥 = −2
Q 1:
−𝑥 2 + 2𝑥 − 7 when 𝑥 = 4
Q 2:
189
9.3b Add, Subtract, Multiply Polynomials – Add
To add polynomials:
To subtract polynomials:
Example 1:
(5𝑥 2 − 7𝑥 + 9) + (2𝑥 2 + 5𝑥 − 14)
Example 2:
(3𝑥 3 − 4𝑥 + 7) − (8𝑥 3 + 9𝑥 − 2)
Q 1:
Q 2:
190
9.3c Add, Subtract, Multiply Polynomials – Multiply Monomial by Polynomial
To multiply a monomial by a polynomial:
Example 1:
Example 2:
2
2
−3𝑥 4 (6𝑥 3 + 2𝑥 − 7)
5𝑥 (6𝑥 − 2𝑥 + 5)
Q 1:
Q 2:
191
9.3d Add, Subtract, Multiply Polynomials – Multiply Binomials
To multiply a binomial by a binomial:
This is often called ___________ which stands for ______________________________________________
Example 1:
Example 2:
(4𝑥 − 2)(5𝑥 + 1)
Q 1:
(3𝑥 − 7)(2𝑥 − 8)
Q 2:
192
9.3e Add, Subtract, Multiply Polynomials – Multiply Trinomials
Multiplying trinomials is just like _____________________ we just have _____________________________
Example 1:
2
(3𝑥 − 4)(9𝑥 + 12𝑥 + 16)
Q 1:
Example 2:
(2𝑥 2 − 6𝑥 + 1)(4𝑥 2 − 2𝑥 − 6)
Q 2:
193
9.3f Add, Subtract, Multiply Polynomials – Multiply Monomials and Binomials
Multiply _____________________ first, then _____________________ the ______________________
Example 1:
Example 2:
4(2𝑥 − 4)(3𝑥 + 1)
Q 1:
3𝑥(𝑥 − 6)(2𝑥 + 5)
Q 2:
194
9.3g Add, Subtract, Multiply Polynomials – Multiply Sum and Difference
(𝑎 + 𝑏)(𝑎 − 𝑏) =
Sum and Difference Shortcut
Example 1:
Example 2:
(𝑥 + 5)(𝑥 − 5)
Q 1:
(5𝑥 − 2)(5𝑥 + 2)
Q 2:
195
9.3h Add, Subtract, Multiply Polynomials – Perfect Squares
(𝑎 + 𝑏)2 =
Notice that (𝑎 + 𝑏)2 is ___________________ 𝑎2 + 𝑏 2 . That is to say, (𝑎 + 𝑏)2 ⎕ 𝑎2 + 𝑏 2
Perfect Square Shortcut:
Example 1:
Example 2:
(𝑥 − 4)2
Q 1:
(2𝑥 + 7)2
Q 2:
You have completed the videos for 9.3 Add, Subtract, Multiply Polynomials. On your own paper
complete the homework assignment.
196
9.4a Polynomial Long Division – Division by Monomials
To divide a polynomial by a monomial we _____________ each ____________ by the __________________
Example 1:
Example 2:
5
4
3𝑥 + 18𝑥 − 9𝑥
3𝑥 2
Q 1:
3
15𝑎6 − 25𝑎5 + 5𝑎4
5𝑎4
Q 2:
197
9.4b Polynomial Long Division – Review Long Division
Long Division Review:
.
5)2632
Example 1
5737
6
Q 1:
198
9.4c Polynomial Long Division – Division by Binomial
Follow the same pattern as __________________________
On the division step focus only on the _________________________
Example 1:
Example 2:
3
2
4𝑥 3 − 6𝑥 2 + 12𝑥 − 5
2𝑥 − 1
𝑥 − 2𝑥 − 15𝑥 + 30
𝑥+4
Q 1:
Q 2:
199
9.4d Polynomial Long Division – Division with Missing Term
The exponents MUST ____________________________
If one is missing we will add ______________
Example 1:
Example 2:
3
2𝑥 3 + 4𝑥 2 + 9
𝑥+3
3𝑥 − 50𝑥 + 4
𝑥−4
Q 1:
Q 2:
You have completed the videos for 9.4 Polynomial Long Division. On your own paper complete the
homework assignment.
Congratulations! You made it through the material for Unit 9: Polynomials. It is time to prepare for
your exam. On a separate sheet of paper complete the practice test. Once you have completed the
practice test, ask your instructor to take the test. Good luck!
200
Unit 10:
Factoring
To work through the unit you should:
1.
2.
3.
4.
5.
6.
7.
Watch a video, as you watch, fill out the workbook (top and example sections).
Complete Q1 and Q2 in WAMAP, put your work in the bottom sections of the page.
Repeat #1 and #2 with each page until you reach
.
Complete the homework assignment on your own paper.
Repeat #1-#4 until you reach the end of the unit.
Complete the practice test on your own paper.
Take the unit exam.
201
10.1a Factor Common Factors and Grouping – Find a GCF
Greatest Common Factor: _____________ factor that __________________ into each term
On variables we use the _________________ exponent
Example 1:
Example 2:
Find the common factor:
15𝑎4 + 10𝑎2
Q 1:
Find the common factor
4𝑎4 𝑏 7 − 12𝑎2 𝑏 6 + 20𝑎𝑏 9
Q 2:
202
10.1b Factor Common Factors and Grouping – Factor a GCF
Factor:
𝑎(𝑏 + 𝑐) =
Put the ________ in front, and divide each ______. What is left goes into the _____________________
Example 1:
Example 2:
9𝑥 4 − 12𝑥 3 + 6𝑥 2
Q 1:
21𝑎4 𝑏 5 − 14𝑎3 𝑏 7 + 7𝑎2 𝑏 4
Q 2:
203
10.1c Factor Common Factors and Grouping – Binomial GCF
The GCF can be a ____________________
Example 1:
Example 2:
5𝑥(2𝑦 − 7) + 6𝑦(2𝑦 − 7)
Q 1:
3𝑥(2𝑥 + 1) − 7(2𝑥 + 1)
Q 2:
204
10.1d Factor Common Factors and Grouping - Grouping
Grouping: GCF of the _____________ and _________________
Then factor out the __________________________ (if it matches)
Example 1:
Example 2:
6𝑥 2 + 3𝑥𝑦 + 2𝑥 + 𝑦
15𝑥𝑦 + 10𝑦 − 18𝑥 − 12
Q 1:
Q 2:
205
10.1e Factor Common Factors and Grouping – Grouping with Change of Order
If the binomials don’t match:
Example 1:
Example 2:
2
6𝑥𝑦 − 20 + 8𝑥 − 15𝑦
12𝑎 − 7𝑏 + 3𝑎𝑏 − 28𝑎
Q 1:
Q 2:
You have completed the videos for 10.1 Factor Common Factors and Grouping. On your own paper
complete the homework assignment.
206
10.2a Factor Trinomials – Reverse FOIL
Recall FOIL: (𝑎 + 𝑏)(𝑐 + 𝑑) =
________________ multiplies to _______________ and _________________ multiplies to _______________
The _________________ and __________________ must add to the ______________________
This may take some ___________________________
Example 1:
Example 2:
2
12𝑥 2 + 16𝑥 − 3
3𝑥 + 11𝑥 + 10
Q 1:
Q 2:
207
10.2b Factor Trinomials – Two Variables
Be aware of _______________ variables when using reverse ____________
Example 1:
Example 2:
2
12𝑥 − 5𝑥𝑦 − 2𝑦
Q 1:
2
6𝑥 2 − 17𝑥𝑦 + 10𝑦 2
Q 2:
208
10.2c Factor Trinomials – With GCF
Always factor the _______ first!
Example 1:
Example 2:
4
3
18𝑥 − 21𝑥 − 15𝑥
Q 1:
2
16𝑥 3 + 28𝑥 2 𝑦 − 30𝑥𝑦 2
Q 2:
209
10.2d Factor Trinomials – Without a Leading Coefficient
If the leading coefficient (in front of 𝑥 2 ) is a 1, then the two numbers will _________ to the ______________
Note: This only works if the leading coefficient is _____
Example 1:
Example 2:
𝑥 2 − 2𝑥 − 8
Q 1:
𝑥 2 + 7𝑥𝑦 − 8𝑦 2
Q 2:
You have completed the videos for 10.2 Factor Trinomials. On your own paper complete the
homework assignment.
210
10.3a Factoring Tricks – Perfect Squares
(𝑎 + 𝑏)2 =
If we can take the square root of the first and last term it _________ be a ______________________
Example 1:
Example 2:
2
9𝑥 2 + 30𝑥𝑦 + 25𝑦 2
𝑥 − 10𝑥 + 25
Q 1:
Q 2:
211
10.3b Factoring Tricks – Difference of Squares
(𝑎 + 𝑏)(𝑎 − 𝑏) =
Difference of Squares:
Example 1:
Example 2:
2
49𝑥 2 − 25𝑦 2
𝑎 − 81
Q 1:
Q 2:
212
10.3c Factoring Tricks – Sum of Squares
Factor: 𝑎2 + 𝑏 2
Sum of squares is always _______________ (this means it ______________ be factored)
Example 1:
Example 2:
𝑥2 + 9
Q 1:
32𝑎2 𝑏 + 50𝑏 3
Q 2:
213
10.3d Factoring Tricks – Sum and Difference of Cubes
Sum of Cubes: 𝑎3 + 𝑏 3 =
Difference of Cubes: 𝑎3 − 𝑏 3 =
Some cubes worth memorizing:
Example 1:
Example 2:
3
8𝑎3 − 27𝑦 3
𝑚 + 125
Q 1:
Q 2:
214
10.3e Factoring Tricks – Difference of 4th Powers
The square root of 𝑥 4 is ______
With fourth powers we can use ______________________ twice!
Example 1:
Example 2:
𝑎4 − 16
Q 1:
81𝑥 4 − 256
Q 2:
215
10.3f Factoring Tricks – Difference of 6th Powers
The square root of 𝑥 6 is _____ and the cubed root of 𝑥 6 is _____
A difference of 6th powers may be a difference of ____________ or a difference of ________________
Use the ___________________ to decide which formula to use
Example 1:
Example 2:
6
𝑥 − 49𝑦
Q 1:
6
8𝑎6 − 27𝑏 6
Q 2:
216
10.3g Factoring Tricks – With GCF
Always factor the _______ first!
Example 1:
Example 2:
3
2𝑥 2 𝑦 − 12𝑥𝑦 + 18𝑦
9𝑥 − 81𝑥
Q 1:
Q 2:
You have completed the videos for 10.3 Factoring Tricks. On your own paper complete the
homework assignment.
217
10.4 Factoring Strategy
Always factor the _______ first!
2 terms:
3 terms:
Example 1:
4 terms:
Example 2:
Which method would you use?
25𝑥 2 − 16
Example 3:
Which method would you use?
𝑥 2 − 𝑥 − 20
Q 1:
Which method would you use?
𝑥𝑦 + 2𝑦 + 5𝑥 + 10
Q 2:
Q 3:
Q 4:
Q 5:
You have completed the videos for 10.4 Factoring Strategy. On your own paper complete the
homework assignment.
218
10.5a Solving Equations by Factoring – Zero Product Rule
Zero Product Rule: if 𝑎𝑏 = 0 then ________________
To solve we set each ___________________ equal to _________________
Example 1:
Example 2:
(5𝑥 − 1)(2𝑥 + 5) = 0
Q 1:
2𝑥(𝑥 − 6)(2𝑥 + 3) = 0
Q 2:
219
10.5b Solving Equations by Factoring – Solve by Factoring
If there is an 𝑥 2 and 𝑥 in the equation, we need to __________________ before we ______________
Example 1:
Example 2:
2
3𝑥 2 + 𝑥 − 4 = 0
𝑥 − 4𝑥 − 12 = 0
Q 1:
Q 2:
220
10.5c Solving Equations by Factoring – Must Equal Zero
Before we factor, the equation must equal _______________
To make factoring easier, we want the ______ term to be ________________________
Example 1:
Example 2:
2
−2𝑥 2 = 𝑥 − 3
5𝑥 = 2𝑥 + 16
Q 1:
Q 2:
221
10.5d Solving Equations by Factoring – Simplify First
Before we make the equation equal zero, we may have to __________________ first
Example 1:
Example 2:
(2𝑥 − 3)(3𝑥 + 1) = −8𝑥 − 1
2𝑥(𝑥 + 4) = 3𝑥 − 3
Q 1:
Q 2:
222
10.5e Solving Equations by Factoring – GCF’s as Factors
When solving do not forget that the _______ is a _________________ also.
If there is no ___________________ in the GCF then we can ____________ it.
Example 1:
Example 2:
3
2
6𝑥 2 = 36 − 15𝑥
4𝑥 − 12𝑥 = 40𝑥
Q 1:
Q 2:
You have completed the videos for 10.5 Solving Equations by Factoring. On your own paper
complete the homework assignment.
Congratulations! You made it through the material for Unit 10: Factoring. It is time to prepare for
your exam. On a separate sheet of paper complete the practice test. Once you have completed the
practice test, ask your instructor to take the test. Good luck!
223
Unit 11:
Rational Expressions
To work through the unit you should:
1.
2.
3.
4.
5.
6.
7.
Watch a video, as you watch, fill out the workbook (top and example sections).
Complete Q1 and Q2 in WAMAP, put your work in the bottom sections of the page.
Repeat #1 and #2 with each page until you reach
.
Complete the homework assignment on your own paper.
Repeat #1-#4 until you reach the end of the unit.
Complete the practice test on your own paper.
Take the unit exam.
224
11.1a Reduce Rational Expressions – Evaluate Rational Expressions
Rational Expression: Quotient of two _____________________
Example 1:
Example 2:
𝑥 2 −2𝑥−8
𝑥−4
Q 1:
𝑥 2 −𝑥−6
when 𝑥 = −4
𝑥 2 +𝑥−12
Q 2:
225
when 𝑥 = 2
11.1b Reduce Rational Expressions – Review Reducing Fractions
To reduce fractions we ___________________ common ____________________
Example 1:
Example 2:
48𝑥 2 𝑦
18𝑥𝑦 3
24
15
Q 1:
Q 2:
226
11.1c Reduce Rational Expressions – Reduce Rational Expressions
To reduce fractions we ___________________ common ____________________
This means we must first _____________________
Example 1:
Example 2:
2
9𝑥 2 − 30𝑥 + 25
9𝑥 2 − 25
2𝑥 + 5𝑥 − 3
2𝑥 2 − 5𝑥 + 2
Q 1:
Q 2:
You have completed the videos for 11.1 Reduce Rational Expressions. On your own paper complete
the homework assignment.
227
11.2a Multiply and Divide Rational Expressions – Review Multiply and Divide Fractions
To multiply we ___________________ common ____________________ then multiply _________________
Division is the same, with one extra step at the start: _________________ by the ___________________
Example 1:
Example 2:
6 21
∙
35 10
Q 1:
5 10
÷
8 3
Q 2:
228
11.2b Multiply and Divide Rational Expressions – Multiply or Divide Rational Expressions
To multiply we ___________________ common ____________________ then multiply _________________
This means we must first _____________________
Division is the same, with one extra step at the start: _________________ by the ___________________
Example 1:
𝑥 2 + 3𝑥 + 2 𝑥 2 − 5𝑥 + 6
∙
4𝑥 − 12
𝑥2 − 4
Q 1:
Example 2:
3𝑥 2 + 5𝑥 − 2
6𝑥 2 + 𝑥 − 1
÷
𝑥 2 + 3𝑥 + 2 2𝑥 3 − 6𝑥 2 − 8𝑥
Q 2:
229
11.2c Multiply and Divide Rational Expressions – Multiply and Divide Rational Expressions
To divide:
To multiply we ___________________ common ____________________ then multiply _________________
This means we must first _____________________
Example 1:
𝑥 2 + 3𝑥 − 10 2𝑥 2 − 𝑥 − 3 8𝑥 + 20
∙
÷
𝑥 2 + 6𝑥 + 5 2𝑥 2 + 𝑥 − 6 6𝑥 + 15
Example 2:
𝑥2 − 1
2𝑥 2 − 𝑥 − 15 2𝑥 2 + 3𝑥 − 5
∙
÷
𝑥 2 − 𝑥 − 6 3𝑥 2 − 𝑥 − 4 3𝑥 2 + 2𝑥 − 8
Q 1:
Q 2:
You have completed the videos for 11.2 Multiply and Divide Rational Expressions. On your own
paper complete the homework assignment.
230
11.3a Add and Subtract Rational Expressions – Review LCD of Numbers with Prime Factorization
Prime Factorization:
To find the LCD use _________ factors with ____________ exponents
Example 1:
Example 2:
Find the LCD:
20 and 36
Q 1:
Find the LCD:
18, 54 and 81
Q 2:
231
11.3b Add and Subtract Rational Expressions – LCD of Monomials
To find the LCD with variables use _________ factors with ____________ exponents
Example 1:
Example 2:
Find the LCD:
5𝑥 3 𝑦 2 and 4𝑥 2 𝑦 5
Q 1:
Find the LCD:
7𝑎𝑏 2 𝑐 and 3𝑎4 𝑏
Q 2:
232
11.3c Add and Subtract Rational Expressions – LCD of Polynomials
To find the LCD with polynomials use _________ factors with ____________ exponents
This means we must first _____________________
Example 1:
Example 2:
Find the LCD:
𝑥 + 3𝑥 − 18 and 𝑥 2 + 4𝑥 − 21
Find the LCD:
𝑥 − 10𝑥 + 25 and 𝑥 2 − 𝑥 − 20
2
Q 1:
2
Q 2:
233
11.3d Add and Subtract Rational Expressions – Review Adding and Subtracting Fractions
To add or subtract we ___________ the denominators by _____________ by the missing ________________
Example 1:
Example 2:
8
3
−
14 10
5
7
+
21 15
Q 1:
Q 2:
234
11.3e Add and Subtract Rational Expressions – Add and Subtract with Common Denominator
Add the __________________ and keep the __________________
When subtracting we will first _________________ the negative
Don’t forget to ______________
Example 1:
Example 2:
𝑥 2 + 4𝑥
𝑥+6
+
𝑥 2 − 2𝑥 − 15 𝑥 2 − 2𝑥 − 15
Q 1:
𝑥 2 + 2𝑥
6𝑥 + 5
−
2𝑥 2 − 9𝑥 − 5 2𝑥 2 − 9𝑥 − 5
Q 2:
235
11.3f Add and Subtract Rational Expressions – Add and Subtract with Different Denominators
To add or subtract we ___________ the denominators by _____________ by the missing ________________
This means we must first _____________________ the denominators
Example 1:
Example 2:
2𝑥
5
+ 2
−9 𝑥 +𝑥−6
𝑥2
𝑥2
Q 1:
2𝑥 + 7
3𝑥 − 2
− 2
− 2𝑥 − 3 𝑥 + 6𝑥 + 5
Q 2:
You have completed the videos for 11.3 Add and Subtract Rational Expressions. On your own paper
complete the homework assignment.
236
Conversion Factors
LENGTH
English
Metric (meter)
12 in = 1 ft
3 ft = 1 yd
1 mi = 5280 ft
1000 mm = 1 m
100 cm = 1 m
10 dm = 1 m
1 dam = 10 m
1 hm = 100 m
1 km = 1000 m
TEMPERATURE
English to Metric
1 in = 2.54 cm
VOLUME
English
Metric (liter)
8 fl oz = 1 cup (c)
2 cups (c) = 1 pint (pt)
2 pints (pt) = 1 quart (qt)
4 quarts (qt) = 1 gallon (gal)
1000 mL = 1 L
100 cL = 1 L
10 dL = 1 L
1 daL = 10 L
1 hL = 100 L
1 kL = 1000 L
1 mL = 1 cc = 1 cm3
TIME
60 seconds (sec) = 1 minute (min)
60 minutes (min) = 1 hour (hr)
24 hours (hr) = 1 day
52 weeks = 1 year
365 days = 1 year
English to Metric
1 gallon (gal) = 3.79 liter (L)
1 in3 = 16.39 mL
WEIGHT (MASS)
English
Metric (gram)
16 oz = 1 pound (lb)
2,000 lb = 1 Ton (T)
1000 mg = 1 g
100 cg = 1 g
10 dg = 1 g
1 dag = 10 g
1 hg = 100 g
1 kg = 1000 g
INTEREST
.
Simple:
Continuous:
A = Pert
Compound:
Annual: n=1
Semiannual: n=2
English to Metric
Quarterly: n=4
2.20 lb = 1 kg
237
.
11.4a Dimensional Analysis – One Step Conversions
57 𝑖𝑛
Consider: (
1
1 𝑓𝑡
) (12𝑖𝑛) =
Divide out units by placing them in the __________________ part of the fraction
Conversion factor: Same _________ in numerator and denominator, but different _________
Dimensional analysis: Multiply by a _____________________ to convert units
Example 1:
Convert 3.8 inches to centimeters
Example 2:
Convert 48 kilograms to pounds
Q 1:
Q 2:
238
11.4b Dimensional Analysis – Multi-Step Conversions
If we do not have the correct conversion factor we can convert using _______________ conversion factors
Example 1:
Example 2:
Convert 5 feet to meters
Q 1:
Convert 3 miles to yards
Q 2:
239
11.4c Dimensional Analysis – Dual Unit Conversions
Dual Unit:
“Per” is the _________________________
With dual units we convert ___________________________
Example 1:
Convert 100 ft per sec to mi per hr
Example 2:
Convert 25 mi per hr to km per min
Q 1:
Q 2:
You have completed the videos for 11.4 Dimensional Analysis. On your own paper complete the
homework assignment.
Congratulations! You made it through the material for Unit 11: Rational Expressions. It is time to
prepare for your exam. On a separate sheet of paper complete the practice test. Once you have
completed the practice test, ask your instructor to take the test. Good luck!
240
Unit 12:
Proficiency Exam #2
To work through this unit you should:
1. Complete the practice tests on your own paper.
2. Take the (two part) unit exam.
241
Unit 13:
Compound Inequalities
To work through the unit you should:
1.
2.
3.
4.
5.
6.
7.
Watch a video, as you watch, fill out the workbook (top and example sections).
Complete Q1 and Q2 in WAMAP, put your work in the bottom sections of the page.
Repeat #1 and #2 with each page until you reach
.
Complete the homework assignment on your own paper.
Repeat #1-#4 until you reach the end of the unit.
Complete the practice test on your own paper.
Take the unit exam.
242
13.1a Review Inequalities – Graph and Interval Notation
Graphing Inequalities: Use ______ for greater/less than and use ______ for “or equal to”
Interval Notation: Use _____ for greater/less than and use ______ for “or equal to”
Example 1:
Graph and give interval notation:
𝑥≥3
Example 2:
Graph and give interval notation:
𝑥 < −2
Q 1:
Q 2:
243
13.1b Review Inequalities – Solve
Inequalities solve just like equations except if we _________________ or ________________ by a
_________________, then we must ___________________________
Example 1:
Example 2:
2𝑥 − 7 > 4𝑥 + 1
Q 1:
5 − 4𝑥 ≥ 13
Q 2:
You have completed the videos for 13.1 Review Inequalities. On your own paper complete the
homework assignment.
244
13.2a Compound Inequalities – OR (two directions)
OR:
First we will _____________ each part above the number line, then we will _____________ the union (OR)
Symbol for Union:
Example 1:
4𝑥 + 7 < −5 OR −4𝑥 − 8 ≤ −20
Example 2:
8𝑥 + 9 < 4𝑥 − 19 OR 2(4𝑥 − 8) − 2 ≤ 12𝑥 − 50
Q 1:
Q 2:
245
13.2b Compound Inequalities – OR (one direction)
With an OR if both graphs go the same direction than we use the _______________________
Example 1:
4𝑥 − 6 > 10 OR 5 − 2𝑥 ≤ 7
Q 1:
Example 2:
3𝑥 + 5 < 2𝑥 − 9 OR 7𝑥 + 3 ≤ 5(𝑥 − 1)
Q 2:
246
13.2c Compound Inequalities – AND (between)
AND:
First we will _____________ each part above the number line, then we will use the _____________ (AND)
Example 1:
6𝑥 + 5 < 11 AND −7𝑥 + 2 ≤ 44
Example 2:
Q 1:
Q 2:
11𝑥 − 10 > 3𝑥 − 2 AND 2(5𝑥 − 3) + 2 ≥ 18𝑥 − 52
247
13.2d Compound Inequalities – AND (one direction)
With an AND if both graphs go the same direction than we use the _______________________
Example 1:
5𝑥 − 6 ≥ 26 AND 3𝑥 + 1 > 𝑥 − 9
Example 2:
2(4𝑥 + 4) > 6𝑥 + 2 AND 7 − 𝑥 ≤ 3 + 𝑥
Q 1:
Q 2:
248
13.2e Compound Inequalities – Special Cases
OR can give us ____________ of number line or ________________________, in interval notation _________
AND can give us ___________ of the number line or _______________________, in interval notation ______
Example 1:
2𝑥 + 1 < 𝑥 − 3 OR 3(𝑥 + 1) ≥ 𝑥 − 15
Example 2:
−3(4𝑥 − 1) ≤ 15 AND 2𝑥 − 3 ≤ −9
Q 1:
Q 2:
You have completed the videos for 13.2 Compound Inequalities. On your own paper complete the
homework assignment.
249
13.3a Absolute Value Inequalities – GreatOR Than
|𝑥| > 2 means the __________ from zero is ____________ than 2.
This is a graph of a compound ______ inequality. It can be written as _______________________
If the absolute value is greatOR than a number we set up an _____
Example 1:
Example 2:
|2𝑥 − 1| ≥ 7
Q 1:
|7𝑥 + 4| > 32
Q 2:
250
13.3b Absolute Value Inequalities – Less Than
|𝑥| < 2 means the __________ from zero is ____________ than 2.
This is a graph of a compound ______ inequality. It can be written as _______________________
If the absolute value is less than a number we set up an _____
Example 1:
Example 2:
|3𝑥 + 7| < 6
Q 1:
|4𝑥 + 1| ≤ 2
Q 2:
251
13.3c Absolute Value Inequalities – Isolate Absolute Value
Before setting up a compound inequality, we must first ________________ the absolute value!
Beware: with absolute value we cannot _____________ or _______________________________
Example 1:
Example 2:
2 − 7|3𝑥 + 4| < −19
Q 1:
5 + 2|4𝑥 − 1| ≤ 17
Q 2:
You have completed the videos for 13.3 Absolute Value Inequalities. On your own paper complete
the homework assignment.
Congratulations! You made it through the material for Unit 13: Compound Inequalities. It is time to
prepare for your exam. On a separate sheet of paper complete the practice test. Once you have
completed the practice test, ask your instructor to take the test. Good luck!
252
Unit 14:
Systems of Equations
To work through the unit you should:
1.
2.
3.
4.
5.
6.
7.
Watch a video, as you watch, fill out the workbook (top and example sections).
Complete Q1 and Q2 in WAMAP, put your work in the bottom sections of the page.
Repeat #1 and #2 with each page until you reach
.
Complete the homework assignment on your own paper.
Repeat #1-#4 until you reach the end of the unit.
Complete the practice test on your own paper.
Take the unit exam.
253
14.1a Systems by Graphing - Solutions
The points on a line are the ___________________ to the equation
The intersection of two lines is the _______________ to both equations!
Other options: ______________ lines have ____ solutions. ______________ lines have __________ solutions
Example 1:
What is the solution for both lines?
Example 2:
What is the solution for both lines?
Example 3:
What is the solution for both lines?
Q 1:
254
14.1b Systems by Graphing – Solve with Intercept Form
To graph lines remember the equation _________________________
Start with the ____________________ or ____ and use the _____________ or _____ to find the next point
Example 1:
Example 2:
2
𝑦 =− 𝑥+3
3
𝑦 = 2𝑥 − 5
Q 1:
2𝑥 − 𝑦 = −4
𝑥+𝑦 =1
Q 2:
You have completed the videos for 14.1 Systems by Graphing. On your own paper complete the
homework assignment.
255
14.2a Systems – Introduction to Substitution
Substitution: Replace the _______________ with what it ____________________
Example 1:
Example 2:
𝑥 = −3
2𝑥 − 3𝑦 = 12
Q 1:
4𝑥 − 7𝑦 = 11
𝑦 = −1
Q 2:
256
14.2b Systems – Substitute an Expression
Just as we can replace a variable with a number, we can also replace it with an ___________________
Whenever we substitute it is important to remember __________________________
Example 1:
Example 2:
𝑦 = 5𝑥 − 3
−𝑥 − 5𝑦 = −11
2𝑥 − 6𝑦 = −24
𝑥 = 5𝑦 − 22
257
Q 1:
Q 2:
258
14.2c Systems – Solve for a Variable
To use substitution we may have to ___________________ a lone variable
If there are several lone variables _________________________
Example 1:
Example 2:
6𝑥 + 4𝑦 = −14
𝑥 − 2𝑦 = −13
−5𝑥 + 𝑦 = −17
7𝑥 + 8𝑦 = 5
259
Q 1:
Q 2:
260
14.2d Systems – Substitution Special Cases
If the variables subtract out to zero then it means either there is _____________________________
or _____________________________
Example 1:
Example 2:
𝑥 + 4𝑦 = −7
21 + 3𝑥 = −12𝑦
5𝑥 + 𝑦 = 3
8 − 3𝑦 = 15𝑥
261
Q 1:
Q 2:
262
14.2e Systems – Addition/Elimination
If there is no lone variable, it may be better to use ______________________
This method works by adding the _______________ and ______________ sides of the equations together
Example 1:
Example 2:
−8𝑥 − 3𝑦 = −12
2𝑥 + 3𝑦 = −6
Q 1:
−5𝑥 + 9𝑦 = 29
5𝑥 − 6𝑦 = −11
Q 2:
263
14.2f Systems – Addition/Elimination and Multiplying an Equation
Addition only works if one of the variables have _______________________________________
To get opposites we can multiply ___________________________ of an equation to get the value we want
Be sure when multiplying to have a ________________ in front of either the ___ or the ____
Example 1:
Example 2:
2𝑥 − 4𝑦 = −4
4𝑥 + 5𝑦 = −21
−5𝑥 + 3𝑦 = −3
−7𝑥 + 12𝑦 = 14
264
Q 1:
Q 2:
265
14.2g Systems – Addition/Elimination and Multiplying Both Equations
Sometimes we may have to multiply ________________________ by something to get opposites
The opposite we look for is the ________ of both coefficients
Example 1:
Example 2:
−6𝑥 + 4𝑦 = 26
4𝑥 − 7𝑦 = −13
3𝑥 + 7𝑦 = 2
10𝑥 + 5𝑦 = −30
266
Q 1:
Q 2:
267
14.2h Systems – Addition/Elimination Special Cases
If the variables subtract out to zero than it means either there is _____________________________
or _____________________________
Example 1:
Example 2:
2𝑥 − 4𝑦 = 16
3𝑥 − 6𝑦 = 20
−10𝑥 + 4𝑦 = −6
25𝑥 − 10𝑦 = 15
268
Q 1:
Q 2:
You have completed the videos for 14.2 Systems. On your own paper complete the homework
assignment.
269
14.3a Systems with Three Variables – Simple
To solve systems with three variables we must _______________ the ____________ variable _____________
This will give us ______ equations with _______ variables we can then solve for!
Example 1:
Example 2:
3𝑥 − 3𝑦 + 5𝑧 = 16
2𝑥 − 6𝑦 − 5𝑧 = 35
−5𝑥 − 12𝑦 + 5𝑧 = 28
−𝑥 + 2𝑦 + 4𝑧 = −20
−2𝑥 − 2𝑦 − 3𝑧 = 5
4𝑥 − 2𝑦 − 2𝑧 = 26
270
Q 1:
Q 2:
271
14.3b Systems with Three Variables – Multiply to Eliminate
To eliminate a variable we may have to __________________ one or more equations to get ______________
Example 1:
−2𝑥 − 2𝑦 + 3𝑧 = −6
3𝑥 − 3𝑦 − 2𝑧 = −17
5𝑥 − 4𝑦 + 5𝑧 = 11
272
Q 1:
You have completed the videos for 14.3 Systems with Three Variables. On your own paper complete
the homework assignment.
273
14.4a Applications of Systems – Value Comparison
Define the ______________________
Make an equation for the ___________________
Make an equation for the ___________________
Example 1:
Brian has twice as many dimes as quarters. If the
value of the coins is $4.95, how many of each does he
have?
Example 2:
A child has three more nickels than dimes in her
piggy-bank. If she has $1.95 in her bank, how many
of each does she have?
274
Q 1:
Q 2:
275
14.4b Applications of Systems – Value with Total
Define the ______________________
Make an equation for the ___________________
Make an equation for the ___________________
Example 1:
Scott has $2.25 in his pocket made up of quarters and
dimes. If there are 12 coins, how many of each coin
does he have?
Example 2:
If 105 people attended a concert and tickets for
adults cost $2.50 while tickets for children cost $1.75
and total receipts for the concert were $228, how
many children and how many adults went to the
concert?
276
Q 1:
Q 2:
277
14.4c Applications of Systems – Interest Comparison
Define the ______________________
Make an equation for the ___________________
Make an equation for the ___________________
Beware: When using a percent we must ___________________
Example 1:
Sophia invested $1900 in one account and $1500 in
another account that paid 3% higher interest rate.
After one year she had earned $113 in interest. At
what rates did she invest?
Example 2:
Carlos invested $2500 in one account and $1000 in
another which paid 4% lower interest. At the end of
a year he had earned $345 in interest. At what rates
did he invest?
278
Q 1:
Q 2:
279
14.4d Applications of Systems – Interest with Total Principle
Define the ______________________
Make an equation for the ___________________
Make an equation for the ___________________
Beware: When using a percent we must ___________________
Example 1:
A woman invests $4600 in two different accounts.
The first paid 13%, the second paid 12% interest. At
the end of the first year she had earned $586 in
interest. How much was in each account?
Example 2:
A bank loaned out $4900 to two different
companies. The first loan had a 4% interest rate; the
second had a 13% interest rate. At the end of the
first year the loan had accrued $421 in interest. How
much was loaned at each rate?
280
Q 1:
Q 2:
281
14.4e Applications of Systems – Mixture with Starting Amount
Define the ______________________
Make an equation for the ___________________
Make an equation for the ___________________
Example 1:
A store owner wants to mix chocolate and nuts to
make a new candy. How many pounds of chocolate
which costs $1.50 per pound should be mixed with 40
pounds of nuts that cost $3.00 per pound to make a
mixture worth $2.50 per pound?
Example 2:
You need a 55% alcohol solution. On hand, you have
600 mL of 10% alcohol mixture. You also have a 95%
alcohol mixture. How much of the 95% mixture
should you add to obtain your desired solution?
282
Q 1:
Q 2:
283
14.4f Applications of Systems – Mixture with Final Amount
Define the ______________________
Make an equation for the ___________________
Make an equation for the ___________________
Example 1:
A chemist needs to create 100 mL of a 38% acid
solution. On hand she has a 20% acid solution and a
50% acid solution. How many mL of each should she
use?
Example 2:
A coffee distributor needs to mix a coffee blend that
normally sells for $8.90 per pound with another
coffee blend that normally sells for $11.16 per
pound, how many pounds of each kind of coffee
should they mix if the distributer needs 50 pounds of
the new mix to sell for $9.85?
284
Q 1:
Q 2:
You have completed the videos for 14.4 Applications of Systems. On your own paper complete the
homework assignment.
Congratulations! You made it through the material for Unit 14: Systems of Equations. It is time to
prepare for your exam. On a separate sheet of paper complete the practice test. Once you have
completed the practice test, ask your instructor to take the test. Good luck!
285
Unit 15:
Radicals
To work through the unit you should:
1.
2.
3.
4.
5.
6.
7.
Watch a video, as you watch, fill out the workbook (top and example sections).
Complete Q1 and Q2 in WAMAP, put your work in the bottom sections of the page.
Repeat #1 and #2 with each page until you reach
.
Complete the homework assignment on your own paper.
Repeat #1-#4 until you reach the end of the unit.
Complete the practice test on your own paper.
Take the unit exam.
286
15.1a Simplify Radicals - Variables
𝑛
Radical: √𝑎 = 𝑏 where ______________. The 𝑛 is called the ________________.
Square Root: √𝑎 = 𝑏 where ______________. The index on a square root is always ____
Radicals divide the ______________ by the __________________
The whole number is how many “things” __________ and the remainder is how many “things” ___________
Example 1:
Example 2:
4
√𝑎3
Q 1:
√𝑏19
Q 2:
287
15.1b Simplify Radicals – Several Variables
Work with ___________ variable at a time
Example 1:
Example 2:
4
√𝑎13 𝑏 23 𝑐10 𝑑 3 𝑒 36
√𝑎5 𝑏 8 𝑐15
Q 1:
Q 2:
288
15.1c Simplify Radicals – Using Prime Factorization
Prime Factorization:
To find a prime factorization we _______________ by _______________
A few prime numbers:
Roots of numbers are difficult, find the ____________________ so that we can divide the ______________
by the __________________
Example 1:
Example 2:
3
√750
Q 1:
9√250𝑥 4 𝑦𝑧 5
Q 2:
289
15.1d Simplify Radicals - Binomials
We can only pull ___________________ (separated by _________________________) out of a radical
If we have ______________ (separated by ____ or _____) we must ______________ first!
Example 1:
Example 2:
3
√100𝑥 2 − 16𝑥 4
Q 1:
√216𝑥 6 − 27𝑥 9
Q 2:
You have completed the videos for 15.1 Simplify Radicals. On your own paper complete the
homework assignment.
290
15.2a Add, Subtract, and Multiply Radicals – Add Like Radicals
Simplify: 2𝑥 − 5𝑦 + 4𝑥 + 2𝑦
Simplify: 2√3 − 5√7 + 4√3 + 2√7
When adding and subtracting radicals we can ____________________________________
Example 1:
Example 2:
3
−4√6 + 2√11 + √11 − 5√6
Q 1:
3
√5 + 3√5 − 8√5 + 2√5
Q 2:
291
15.2b Add, Subtract, and Multiply Radicals – Add with Simplifying
Before adding radicals together _______________________
Example 1:
5√50𝑥 + 5√27 − 3√2𝑥 − 2√108
Example 2:
3
3
3
√81𝑥 3 𝑦 − 3𝑦 √32𝑥 2 + 𝑥 3√24𝑦 − √500𝑥 2 𝑦 3
Q 1:
Q 2:
292
15.2c Add, Subtract, and Multiply Radicals – Multiply Monomial Radical Expressions
𝑛
𝑛
Product Rule: 𝑎 √𝑏 ∙ 𝑐 √𝑑 =
Always be sure your final answer is ______________________
Example 1:
Example 2:
4
Q 1:
4
−3√8 ∙ 7 √10
4√6 ∙ 2√15
Q 2:
293
15.2d Add, Subtract, and Multiply Radicals – Multiply Monomial by Binomial Radical Expressions
Recall: 𝑎(𝑏 + 𝑐) =
Always be sure your final answer is ______________________
Example 1:
Example 2:
7√3(√6 + 9)
5√10(2√6 − 3√15)
Q 1:
Q 2:
294
15.2e Add, Subtract, and Multiply Radicals – Multiply Binomial Radical Expressions
Recall: (𝑎 + 𝑏)(𝑐 + 𝑑) =
Always be sure your final answer is ______________________
Example 1:
Example 2:
3
Q 1:
3
(2√9 + 5)(4√3 − 1)
(3√7 − 2√5)(√7 + 6√5)
Q 2:
295
15.2f Add, Subtract, and Multiply Radicals – Square Binomial Radical Expression
Recall: (𝑎 + 𝑏)2 =
Always be sure your final answer is ______________________
Example 1:
Example 2:
(√6 − √2)
Q 1:
2
(2 + 3√7)
Q 2:
296
2
15.2g Add, Subtract, and Multiply Radicals – Multiply Conjugates
Recall: (𝑎 + 𝑏)(𝑎 − 𝑏) =
Always be sure your final answer is ______________________
Example 1:
Example 2:
(4 + 2√7)(4 − 2√7)
Q 1:
(2√3 − √6)(2√3 + √6)
Q 2:
You have completed the videos for 15.2 Add, Subtract, and Multiply Radicals. On your own paper
complete the homework assignment.
297
15.3a Rationalize Denominators – Simplifying with Radicals
Expression with radicals: Always ______________ the _________________ first
Before _____________________ with fractions, be sure to _____________ first
Example 1:
Example 2:
8 − √48
6
15 + √175
10
Q 1:
Q 2:
298
15.3b Rationalize Denominators – Quotient Rule
𝑎
Quotient Rule: √𝑏 =
It may be helpful to reduce the _________________ first and the ______________ second
Example 1:
Example 2:
√48
225𝑥 7
√
20𝑥 3
√150
Q 1:
Q 2:
299
15.3c Rationalize Denominators – Rationalize Monomial Roots in the Denominator
Rationalize Denominators: Never leave a _________________ in the ______________________
To clear radicals: ________________ by extra needed factors in denominator (same in numerator!)
It may be helpful to _________________ first
Hint: _____________ numbers!
Example 1:
Example 2:
5
3
7
√
√𝑏 2
Q 1:
Q 2:
300
7
9𝑎2 𝑏
15.3d Rationalize Denominators – Rationalize Binomial Denominators
What does not work:
1
2+√3
=
Recall: (2 + √3)(
)=
Multiply by the __________________________
Example 1:
Q 1:
Example 2:
6
3 − 5√2
5 − √3
4 + 2√2
Q 2:
You have completed the videos for 15.3 Rationalize Denominators. On your own paper complete
the homework assignment.
301
15.4a Rational Exponents – Convert
𝑛
If we divide the exponent by the index, then √𝑎𝑚 =
The index is the _______________________
Example 1:
Example 2:
Write as an exponent:
Write as a radical: (𝑎𝑏)2/3
7
√𝑚5
Example 3:
Example 4:
Write as a radical: 𝑥 −4/5
Write as an exponent:
1
3
( √5𝑥 )
Q 1:
Q 2:
Q 3:
Q 4:
302
2
15.4b Rational Exponents – Evaluate
To evaluate a rational exponent ______________ to a ________________________
Example 1:
Example 2:
2/5
Evaluate: 27−4/3
Evaluate: 32
Q 1:
Q 2:
303
15.4c Rational Exponents – Simplify
Recall Exponent Properties:
𝑎𝑚 𝑎𝑛 =
𝑎𝑚
=
𝑎𝑛
(𝑎𝑏)𝑚 =
𝑎 𝑚
( ) =
𝑏
(𝑎𝑚 )𝑛 =
𝑎0 =
𝑎−𝑚 =
1
𝑎−𝑚
=
𝑎 −𝑚
( )
=
𝑏
To Simplify:
Example 1:
Example 2:
𝑥
4/3 2/7 5/4 3/7
−1/8
𝑦 𝑥 𝑦
𝑥1/2 𝑦 6/7
256𝑥 3/2 𝑦 −1/3
( 1/4 3/2 −5/2 )
𝑥 𝑦 𝑥
304
Q 1:
Q 2:
You have completed the videos for 15.4 Rational Exponents. On your own paper complete the
homework assignment.
305
15.5a Radicals of Mixed Index – Reduce Index
8
Using rational exponents: √𝑥 6 𝑦 2 =
To reduce the index ____________ the ___________ and the ________________ by the _______
8
Without using rational exponents: √𝑥 6 𝑦 2 =
Hint: ______________ any numbers
Example 1:
Example 2:
15
25
√𝑥 3 𝑦 9 𝑧 6
Q 1:
√32𝑎10 𝑏 5 𝑐 20
Q 2:
306
15.5b Radicals of Mixed Index – Multiply Mixed Index
3
4
Using rational exponents: √𝑎2 𝑏 ∙ √𝑎𝑏 2 =
Get a ____________________ by ________________ the _______________ and ________________
3
4
Without using rational exponents: √𝑎2 𝑏 ∙ √𝑎𝑏 2 =
Hint: __________________ any numbers
Always be sure your final answer is ___________________
Example 1:
Example 2:
4
√𝑚3 𝑛2 𝑝
Q 1:
∙
6
3
√𝑚𝑛2 𝑝3
5
√4𝑥 2 𝑦 ∙ √8𝑥 4 𝑦 2
Q 2:
307
15.5d Radicals of Mixed Index – Divide Mixed Index
Division with mixed index – get a __________________________
Hint: __________________ any numbers
May have to _____________ the denominator (cannot be under a ___________ and under a ____________)
Example 1:
Example 2:
4
√𝑎𝑏 3
√2𝑥 3 𝑦 2
3
6
√𝑎𝑏 2
Q 1:
√32𝑦 4
Q 2:
You have completed the videos for 15.5 Radicals of Mixed Index. On your own paper complete the
homework assignment.
308
15.6a Complex Numbers – Square Roots of Negatives
Define: √−1 =
and therefore 𝑖 2 =
Now we can calculate √−25 =
Expressions with radicals: Always ______________ the _________________ first
Example 1:
Example 2:
√−45
Q 1:
√−6 ∙ √−10
Q 2:
309
15.6b Complex Numbers – Simplify Square Roots of Negatives
Before _____________________ with fractions, be sure to _____________ first
Example 1:
Example 2:
15 + √−300
5
Q 1:
20 + √−80
8
Q 2:
310
15.6c Complex Numbers – Add and Subtract
𝑖 works just like ____________________
This means we can _________________________________
Example 1:
Example 2:
(5 − 3𝑖) + (6 + 𝑖)
Q 1:
(−5 − 2𝑖) − (3 − 6𝑖)
Q 2:
311
15.6d Complex Numbers – Multiply
𝑖 works just like ____________________
Remember 𝑖 2 =
Example 1:
Example 2:
(−3𝑖)(6𝑖)
Example 3:
2𝑖(5 − 2𝑖)
Example 4:
(3 + 2𝑖)2
(4 − 3𝑖)(2 − 5𝑖)
Q 1:
Q 2:
Q 3:
Q 4:
312
15.6e Complex Numbers – Rationalize Monomial Denominators
If 𝑖 =
then we can rationalize it by just multiplying by _____
Example 1:
Example 2:
2−𝑖
−3𝑖
5 + 3𝑖
4𝑖
Q 1:
Q 2:
313
15.6f Complex Numbers – Rationalize Binomial Denominators
Similar to other radicals we can rationalize a binomial by multiplying by the _____________________
(𝑎 + 𝑏𝑖)(𝑎 − 𝑏𝑖) =
Example 1:
Example 2:
4𝑖
2 − 5𝑖
Q 1:
4 − 2𝑖
3 + 5𝑖
Q 2:
You have completed the videos for 15.6 Complex Numbers. On your own paper complete the
homework assignment.
Congratulations! You made it through the material for Unit 15: Radicals. It is time to prepare for
your exam. On a separate sheet of paper complete the practice test. Once you have completed the
practice test, ask your instructor to take the test. Good luck!
314
Unit 16:
Quadratics, Rational Equations,
and Applications
To work through the unit you should:
1.
2.
3.
4.
5.
6.
7.
Watch a video, as you watch, fill out the workbook (top and example sections).
Complete Q1 and Q2 in WAMAP, put your work in the bottom sections of the page.
Repeat #1 and #2 with each page until you reach
.
Complete the homework assignment on your own paper.
Repeat #1-#4 until you reach the end of the unit.
Complete the practice test on your own paper.
Take the unit exam.
315
16.1a Complete the Square – Find c
𝑎2 + 2𝑎𝑏 + 𝑏 2 is easily factored to ______________________
To make 𝑥 2 + 𝑏𝑥 + 𝑐 a perfect square, 𝑐 =
Example 1:
Find 𝑐 and factor the perfect square:
𝑥 2 + 10𝑥 + 𝑐
Example 2:
Find 𝑐 and factor the perfect square
𝑥 2 − 7𝑥 + 𝑐
Example 3:
Find 𝑐 and factor the perfect square:
3
𝑥2 − 𝑥 + 𝑐
7
Example 4:
Find 𝑐 and factor the perfect square:
6
𝑥2 + 𝑥 + 𝑐
5
Q 1:
Q 2:
Q 3:
Q 4:
316
16.1b Complete the Square – Rational Solutions
If 𝑥 2 = 9 then there are ____ solutions for 𝑥, ______ and ________. We can write this as ___________
To complete the square on 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0
1. Separate ________________ and __________________
2. Divide by _______ (everything)
3. Find the _________ and _________ to ________________________
Example 1:
Example 2:
𝑥2 − 𝑥 − 6 = 0
3𝑥 2 = 15𝑥 − 18
317
Q 1:
Q 2:
318
16.1c Complete the Square – Irrational and Complex Solutions
If we can’t simplify the _______________ we _____________________ what we can.
Example 1:
Example 2:
2
8𝑥 + 28 = 4𝑥 2
5𝑥 − 3𝑥 + 2 = 0
319
Q 1:
Q 2:
You have completed the videos for 16.1 Complete the Square. On your own paper complete the
homework assignment.
320
16.2a Quadratic Formula – Finding the Formula
Solve by Completing the Square:
𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0
(How we found the formula is useful to know for the test!)
321
16.2b Quadratic Formula – Using the Formula
If 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 the 𝑥 =
Example 1:
Example 2:
2
5𝑥 2 − 𝑥 + 2 = 0
6𝑥 + 7𝑥 − 3 = 0
Q 1:
Q 2:
322
16.2c Quadratic Formula – Make Equation Equal Zero
Before using the quadratic formula, the equation must equal ______ and be in _____________
That is the equation should look like:
Example 1:
Example 2:
2
3𝑥 2 + 5𝑥 + 2 = 7
2𝑥 = 15 − 7𝑥
Q 1:
Q 2:
323
16.2d Quadratic Formula – Missing Terms
If a term is missing, we use _____ in the quadratic formula
Example 1:
Example 2:
2
5𝑥 2 = 2𝑥
3𝑥 + 54 = 0
Q 1:
Q 2:
You have completed the videos for 16.2 Quadratic Formula. On your own paper complete the
homework assignment.
324
16.3a Equations with Radicals – Odd Roots
The opposite of taking a root is to do an ________________
3
√𝑥 = 4 then 𝑥 =
(Note: This only works for an _________ index)
Example 1:
Example 2:
3
5
√4𝑥 − 7 = 2
√2𝑥 − 5 = 6
Q 1:
Q 2:
325
16.3b Equations with Radicals – Even Roots
The opposite of taking a root is to do an ________________
With even roots we must ___________ the answer in the original equation! (called ____________________)
Recall: (𝑎 + 𝑏)2 =
Example 1:
Example 2:
𝑥 = √5𝑥 + 24
Q 1:
√40 − 3𝑥 = 2𝑥 − 5
Q 2:
326
16.3c Equations with Radicals – Isolate Radical
IMPORTANT: Before we can clear a radical it must first be _________________
Example 1:
Example 2:
4 + 2√2𝑥 − 1 = 2𝑥
2√5𝑥 + 1 − 2 = 2𝑥
327
Q 1:
Q 2:
You have completed the videos for 16.3 Equations with Radicals. On your own paper complete the
homework assignment.
328
16.4a Equations with Exponents – Odd Exponents
The opposite of taking an exponent is to do a ________________
If 𝑥 3 = 8, then 𝑥 =
(Note: This only works for an _________ exponent)
Example 1:
Example 2:
(2𝑥 − 1)3 = 64
(3𝑥 + 5)5 = 32
Q 1:
Q 2:
329
16.4b Equations with Exponents – Even Exponents
Consider: (5)2 =
and (−5)2 =
When we clear an even exponent we have _____________________________
Example 1:
Example 2:
(3𝑥 + 2)4 = 81
(5𝑥 − 1)2 = 49
Q 1:
Q 2:
330
16.4c Equations with Exponents – Isolate Exponent
IMPORTANT: Before we can clear an exponent it must first be _________________
Example 1:
Example 2:
2
5(3𝑥 − 2)2 + 6 = 46
4 − 2(2𝑥 + 1) = −46
Q 1:
Q 2:
331
16.4d Equations with Exponents – Rational Exponents
To multiply to one:
𝑎
𝑏
∙(
)=1
We clear a rational exponent by using a _______________________
Recall 𝑎𝑚/𝑛 =
Recall: Check if original rational exponent has _______________________
Recall: Two solutions if original rational exponent has ___________________
Example 1:
Example 2:
(3𝑥 − 6)
Q 1:
3/2
(5𝑥 + 1)4/5 = 16
= 64
Q 2:
You have completed the videos for 16.4 Equations with Exponents. On your own paper complete
the homework assignment.
332
16.5a Rectangle Problems – Area Problems
Area of a rectangle:
To help visualize the rectangle, __________________________________________
There are three ways to solve any quadratic equation
1.
2.
3.
Example 1:
The length of a rectangle is 2 ft longer than the width.
The area of the rectangle is 48 ft2. What are the
dimensions of the rectangle?
Example 2:
The area of a rectangle is 72 cm2. If the width is 6 cm
less than the length, what are the dimensions of the
rectangle?
Q 1:
Q 2:
333
16.5b Rectangle Problems – Perimeter Problems
Perimeter of a rectangle:
Tip: Solve the ________________ equation for a variable and ________________ in the _________ equation.
Example 1:
The area of a rectangle is 54m2. If the perimeter is 30
meters, what are the dimensions of the rectangle?
Example 2:
The perimeter of a rectangle is 22 inches. If the area
of the same rectangle is 24 in2, what are the
dimensions?
Q 1:
Q 2:
334
16.5c Rectangle Problems – Bigger
We may have to draw _____________ rectangles
Multiply/Add to the _________ to make it equal the ____________ rectangle
Example 1:
Each side of a square is decreased 6 inches. When this
happens, the area of the larger square is 16 times the
area of the smaller square. How many inches is the
side of the original square?
Example 2:
The length of a rectangle is 9 feet longer than it is
wide. If each side is increased 9 feet, then the area is
multiplied by 3. What are the dimensions of the
original rectangle?
Q 1:
Q 2:
335
16.5d Rectangle Problems – Frames
To help visualize the frame __________________________
Remember the frame is on the __________ and ______________, also the ___________ and _____________
Example 1:
A frame measures 13 inches by 10 inches and is of
uniform width. If the area of the picture inside is 54
square inches, what is the width of the frame?
Example 2:
An 8 inch by 12 inch drawing has a frame of uniform
width around it. The area of the frame is equal to the
area of the picture. What is the width of the frame?
Q 1:
Q 2:
336
16.5e Rectangle Problems – Percent of a Field
Clearly identify the area of the ___________________ and ______________________ rectangles!
Be careful with ___________________, is it talking about the ___________, ______________, or _________?
Example 1:
A man mows his 40 ft by 50 ft rectangular lawn in a
spiral pattern starting from the outside edge. By noon
he is 90% done. How wide of a strip has he cut around
the outside edge?
Example 2:
A woman has a 50 ft by 25 ft rectangular field that
he wants to increase by 68% by cultivating a strip of
uniform width around the current field. How wide of
a strip should she cultivate?
Q 1:
Q 2:
You have completed the videos for 16.5 Rectangle Problems. On your own paper complete the
homework assignment.
337
16.6a Rational Equations – Clear Denominator
Recall:
3
4
1
5
2
6
𝑥− =
Clear fractions by multiplying each _________________ by the ________________
Example 1:
Example 2:
4
−2
+𝑥 =
𝑥+5
𝑥+5
5
3
=
−4
𝑥 7𝑥
Q 1:
Q 2:
338
16.6b Rational Equations – Factoring Denominator
To identify all the factors in the ________ we may have to ______________ the ____________________
Example 1:
Example 2:
𝑥
1
−3𝑥 − 8
+
= 2
𝑥 − 6 𝑥 − 7 𝑥 − 13𝑥 + 42
Q 1:
2
9𝑥
1
− 2
=
𝑥+3 𝑥 −9 𝑥−3
Q 2:
339
16.6c Rational Equations – Extraneous Solutions
Because we are working with fractions, the _____________ cannot be _____________
Example 1:
Example 2:
𝑥
2
4𝑥 − 12
+
= 2
𝑥 − 2 𝑥 − 4 𝑥 − 6𝑥 + 8
𝑥
2
−3𝑥 + 56
−
= 2
𝑥 − 8 𝑥 − 4 𝑥 − 12𝑥 + 32
Q 1:
Q 2:
You have completed the videos for 16.6 Rational Equations. On your own paper complete the
homework assignment.
340
16.7a Work Problems – One Unknown Time
Adam does a job in 4 hours. Each hour he does ______ of the job.
Betty does a job in 12 hours. Each hour she does ______ of the job.
Together, each hour they do _____________________ of the job
This means together it would take them _________ hours to do the entire job.
Work equation:
Example 1:
Catherine can paint a house in 15 hours. Dan can
paint it in 30 hours. How long will it take them
working together?
Example 2:
Even can clean a room in 3 hours. If his sister Faith
2
helps, it takes them 2 5 hours. How long will it take
Faith working alone?
Q 1:
Q 2:
341
16.7b Work Problems – Two Unknown Times
Be sure to clearly identify who is the _________________
Example 1:
Tony does a job in 16 hours less time than Marissa,
and they can do it together in 15 hours. How long will
it take each to do the job alone?
Example 2:
Alex can complete his project in 21 hours less than
Hillary. If they work together it can get done in 10
hours. How long does it take each working alone?
Q 1:
Q 2:
You have completed the videos for 16.7 Work Problems. On your own paper complete the
homework assignment.
342
16.8a Distance and Revenue Problems – Simultaneous Products
Simultaneous product: ________ equations with _________ variables that are __________________
To solve: ________________ both by the same ______________________. Then ____________________.
Example 1:
𝑥𝑦 = 72
(𝑥 − 5)(𝑦 + 2) = 56
Q 1:
343
16.8b Distance and Revenue Problems – Revenue
Revenue Equation:
Beware: Profit =
To solve: Divide by what we _______________
Example 1:
A group of college students bought a couch for $80.
However, five of them failed to pay their share so the
others had to each pay $8 more. How many students
were in the original group?
Example 2:
A merchant bought several pieces of silk for $70. He
sold all but two of them at a profit of $4 per piece.
His total profit was $18. How many pieces did he
originally purchase?
344
Q 1:
Q 2:
345
16.8c Distance and Revenue Problems – Distance
Distance Equation:
To solve: Divide by what we _______________
Example 1:
A man rode his bike to a park 60 miles away. On the
return trip he went 2 mph slower which made the trip
take 1 hour longer. How fast did he ride to the park?
Example 2:
After driving through a construction zone for 45
miles, a woman realized that if she had just driven 6
mph faster she would have arrived 2 hours sooner.
How fast did she drive?
346
Q 1:
Q 2:
347
16.8d Distance and Revenue Problems – Streams and Wind
Downwind/stream:
Upwind/stream:
Example 1:
Zoe rows a boat downstream for 80 miles. The return
trip upstream took 12 hours longer. If the current
flows at 3 mph, how fast does Zoe row in still water?
Example 2:
Darius flies a plane against a headwind for 5084
miles. The return trip with the wind took 20 hours
less time. If the wind speed is 10 mph, how fast does
Darius fly the plane when there is no wind?
348
Q 1:
Q 2:
You have completed the videos for 16.8 Distance and Revenue Problems. On your own paper
complete the homework assignment.
349
16.9a Compound Fractions – Numbers
Compound/Complex Fractions:
Clear ___________________ by multiplying each _________ by the _______________ of everything
Example 1:
Example 2:
1
2+2
9
1+4
3 5
4+6
1 4
2−3
Q 1:
Q 2:
350
16.9b Compound Fractions – Monomials
Recall: To find the LCD with variables, use the __________________ exponents
Be sure to check for _______________________ by __________________ the numerator and denominator
Example 1:
Example 2:
9
𝑥2
1 3
𝑥 + 𝑥2
1
1
−
𝑦3 𝑥3
1
1
−
𝑥2𝑦3 𝑥3𝑦2
1−
Q 1:
Q 2:
351
16.9c Compound Fractions – Binomials
The LCD could have one or more _______________ in it!
Example 1:
Example 2:
5
𝑥−2
2
3+𝑥−2
Q 1:
𝑥
5
+
𝑥−9 𝑥+9
𝑥
5
𝑥+9−𝑥−9
Q 2:
352
16.9d Compound Fractions – Negative Exponents
Recall: 5𝑥 −3 =
If there is any ______ or __________ we can’t just _________ terms. Instead make ___________________
Example 1:
Example 2:
1 + 10𝑥 −1 + 25𝑥 −2
1 − 25𝑥 −2
Q 1:
8𝑏 −3 + 27𝑎−3
4𝑎−1 𝑏 −3 − 6𝑎 −2 𝑏 −2 + 9𝑎 −3 𝑏 −1
Q 2:
You have completed the videos for 16.9 Compound Fractions. On your own paper complete the
homework assignment.
Congratulations! You made it through the material for Unit 16: Quadratics, Rational Equations, and
Applications. It is time to prepare for your exam. On a separate sheet of paper complete the practice
test. Once you have completed the practice test, ask your instructor to take the test. Good luck!
353
Unit 17:
Functions
To work through the unit you should:
1.
2.
3.
4.
5.
6.
7.
Watch a video, as you watch, fill out the workbook (top and example sections).
Complete Q1 and Q2 in WAMAP, put your work in the bottom sections of the page.
Repeat #1 and #2 with each page until you reach
.
Complete the homework assignment on your own paper.
Repeat #1-#4 until you reach the end of the unit.
Complete the practice test on your own paper.
Take the unit exam.
354
17.1a Evaluate Functions – Functions
Function:
If it is a function we often write ____ which is read __________________________
A graph is a function if it passes the ______________________________, or each ____ has at most one ____
Example 1:
Example 2:
Is the graph a function?
Q 1:
Is the graph a function?
Q 2:
355
17.1b Evaluate Functions – Function Notation
Function notation:
What is inside of the function ___________________ the ________________________
Example 1:
Example 2:
2
𝑓(𝑥) = −𝑥 + 2𝑥 − 5
Find 𝑓(3)
Q 1:
𝑔(𝑥) = √2𝑥 + 5
Find 𝑔(20)
Q 2:
356
17.1c Evaluate Functions – Evaluate Function at an Expression
When replacing a variable we always use _________________________
What is inside of the function ___________________ the ________________________
Example 1:
Example 2:
𝑝(𝑛) = 𝑛2 − 2𝑛 + 5
Find 𝑝(𝑛 − 3)
𝑓(𝑥) = √2𝑥 + 3𝑥
Find 𝑓(8𝑥 2 )
Q 1:
Q 2:
357
17.1d Evaluate Functions – Domain
Domain:
Fractions:
Even Radicals:
Example 1:
Example 2:
Find the domain:
4
𝑓(𝑥) = 3 √2𝑥 − 6 + 4
Example 3:
Find the domain:
𝑔(𝑥) = 3|2𝑥 + 7|2 − 4
Q 1:
Find the domain:
𝑥−1
ℎ(𝑥) = 2
𝑥 −𝑥−2
Q 2:
Q 3:
You have completed the videos for 17.1 Evaluate Functions. On your own paper complete the
homework assignment.
358
17.2a Operations on Functions – Add Functions
Add Functions: (𝑓 + 𝑔)(𝑥) =
With a number we will ________________ both, then ______________ the results
With a variable we will ____________ the two functions __________________. Use ___________________!
Example 1:
Example 2:
𝑓(𝑥) = 𝑥 2 − 5𝑥
𝑔(𝑥) = 𝑥 − 5
Find (𝑓 + 𝑔)(𝑥)
𝑓(𝑥) = 𝑥 − 4
𝑔(𝑥) = 𝑥 2 − 6𝑥 + 8
Find (𝑓 + 𝑔)(−2)
Q 1:
Q 2:
359
17.2b Operations on Functions – Subtract Functions
Subtract Functions: (𝑓 − 𝑔)(𝑥) =
With a number we will ________________ both, then ______________ the results
With a variable we will ____________ the two functions __________________. Use ___________________!
Example 1:
Example 2:
𝑓(𝑥) = 𝑥 2 − 5𝑥
𝑔(𝑥) = 𝑥 − 5
Find (𝑓 − 𝑔)(𝑥)
𝑓(𝑥) = 𝑥 − 4
𝑔(𝑥) = 𝑥 2 − 6𝑥 + 8
Find (𝑓 − 𝑔)(−2)
Q 1:
Q 2:
360
17.2c Operations on Functions – Multiply Functions
Multiply Functions: (𝑓 ∙ 𝑔)(𝑥) =
With a number we will ________________ both, then ______________ the results
With a variable we will ____________ the two functions __________________. Use ___________________!
Example 1:
Example 2:
𝑓(𝑥) = 𝑥 2 − 5𝑥
𝑔(𝑥) = 𝑥 − 5
Find (𝑓 ∙ 𝑔)(𝑥)
𝑓(𝑥) = 𝑥 − 4
𝑔(𝑥) = 𝑥 2 − 6𝑥 + 8
Find (𝑓 ∙ 𝑔)(−2)
Q 1:
Q 2:
361
17.2d Operations on Functions – Divide Functions
𝑓
Divide Functions: (𝑔) (𝑥) =
With a number we will ________________ both, then ______________ the results
With a variable we will ____________ the two functions __________________. Use ___________________!
Beware of ______________ of fractions, the ________________ cannot be ____________
Example 1:
Example 2:
𝑓(𝑥) = 𝑥 2 − 5𝑥
𝑔(𝑥) = 𝑥 − 5
𝑓
Find (𝑔) (𝑥)
𝑓(𝑥) = 𝑥 − 4
𝑔(𝑥) = 𝑥 2 − 6𝑥 + 8
𝑓
Find (𝑔) (−2)
Q 1:
Q 2:
362
17.2e Operations on Functions – Composition of Functions
Composition of Functions:
(𝑓 ∘ 𝑔)(𝑥) =
With numbers, _________________ the __________________ and put ______________ in ______________
With a variable, put the _________________ in for the ___________________ in the __________________
Example 1:
Example 2:
𝑝(𝑥) = 𝑥 2 + 2𝑥
𝑟(𝑥) = 𝑥 + 3
𝑓(𝑥) = √𝑥 + 6
𝑔(𝑥) = 𝑥 + 3
(𝑓 ∘ 𝑔)(7) =
(𝑝 ∘ 𝑟)(𝑥) =
𝑔[𝑓(7)] =
𝑟[𝑝(𝑛)] =
363
Q 1:
Q 2:
364
17.2f Operations on Functions – Compose a Function with Itself
A function can be composed with _____________
Example 1:
Example 2:
𝑔(𝑥) = 𝑥 2 − 3𝑥
Find 𝑔[𝑔(𝑥)]
𝑓(𝑥) = 2𝑥 − 4
Find (𝑓 ∘ 𝑓)(−2)
Q 1:
Q 2:
365
17.2g Operations on Functions – Composition of Several Functions
If we are composing several function, start in the _______________ and work __________
Example 1:
Example 2:
𝑓(𝑥) = 𝑥 + 2
𝑔(𝑥) = 𝑥 2 − 5
ℎ(𝑥) = √3𝑥
Find (𝑓 ∘ 𝑔 ∘ ℎ)(2)
Q 1:
𝑓(𝑥) = 𝑥 + 2
𝑔(𝑥) = 𝑥 2 − 5
ℎ(𝑥) = √3𝑥
Find (𝑓 ∘ 𝑔 ∘ ℎ)(𝑎)
Q 2:
You have completed the videos for 17.2 Operations on Functions. On your own paper complete the
homework assignment.
366
17.3a Inverse Functions – Show Functions are Inverses
Inverse Function:
To test if functions are inverses, calculate __________ and ___________, the answer to both should be ___
Example 1:
Example 2:
Are they inverses?
𝑓(𝑥) = 3𝑥 − 8
𝑥
𝑔(𝑥) = + 8
3
Q 1:
Are they inverses?
5
𝑓(𝑥) =
+6
𝑥−3
5
𝑔(𝑥) =
+3
𝑥−6
Q 2:
367
17.3b Inverse Functions – Finding an Inverse Function
To find an inverse function _______________ the ____ and ____ , then solve for ____.
(the ____ is the y!)
Example 1:
Example 2:
Find the inverse:
−3
ℎ(𝑥) =
−2
𝑥−1
Find the inverse:
3
𝑔(𝑥) = 5√𝑥 − 6 + 4
368
Q 1:
Q 2:
369
17.3c Inverse Functions – Inverse of Rational Functions
Clear fractions by ________________________________
Put the terms with ____ on one side and ____________________ on the other side
Factor out the _____ and _________________ to get it alone
Example 1:
Example 2:
Find the inverse:
2𝑥 − 5
𝑓(𝑥) =
𝑥+3
Find the inverse:
5𝑥 + 1
𝑔(𝑥) =
2𝑥 − 5
370
Q 1:
Q 2:
You have completed the videos for 17.3 Inverse Functions. On your own paper complete the
homework assignment.
371
17.4 Graphs of Quadratic Functions – Key Points
Quadratic Equations are of the form:
Quadratic Graph:
Example 1:
Key points:
Example 2:
Graph the function:
𝑓(𝑥) = 𝑥 2 − 2𝑥 − 3
Graph the function
𝑓(𝑥) = −3𝑥 2 + 12𝑥 − 9
372
Q 1:
Q 2:
You have completed the videos for 17.4 Graphs of Quadratic Functions. On your own paper
complete the homework assignment.
373
17.5a Exponential Equations – With Common Base
Exponential functions:
Solving exponential functions: If the ______________ are equal, then the ___________________ are equal
Example 1:
Example 2:
3𝑥−6
7
Q 1:
5𝑥+2
45−𝑥 = 43𝑥
=7
Q 2:
374
17.5b Exponential Equations – Find a Common Base
If we don’t have a common base, then we find the _____________________ of the base
Recall exponent property: (𝑎𝑚 )𝑛 =
When using the above property we may have to ________________________
Example 1:
Example 2:
2𝑥
27
Q 1:
82𝑥−4 = 16𝑥+3
=9
Q 2:
375
17.5c Exponential Equations – With Negative Exponents
Fractions are created by _____________________________
Example 1:
𝑥
Example 2:
1 3𝑥−1
( )
= 1254𝑥+2
25
1
( ) = 814𝑥
3
Q 1:
Q 2:
You have completed the videos for 17.5 Exponential Equations. On your own paper complete the
homework assignment.
376
17.6a Compound Interest – N Compound a Year
Compound interest:
𝑟 𝑛𝑡
𝑛 compounds per year: 𝐴 = 𝑃 (1 + 𝑛)
𝐴=
𝑃=
𝑟=
𝑛=
𝑡=
Example 1:
Suppose you invest $13,000 in an account that pays
8% interest compounded monthly. How much would
be in the account after 9 years?
Example 2:
A bank loans out $800 at 3% interest compounded
quarterly. If the loan is paid in full after five years,
what is the balance owed?
Q 1:
Q 2:
377
17.6b Compound Interest – Continuous Interest
Continuous interest:
𝐴 = 𝑃𝑒 𝑟𝑡
𝐴=
𝑃=
𝑒=
𝑟=
𝑡=
Example 1:
An investment of $25,000 is at an interest rate of
11.5% compounded continuously. What is the
balance after 20 years?
Example 2:
What is the balance at the end of 10 years on an
investment of $13,000 at 4% compounded
continuously?
Q 1:
Q 2:
You have completed the videos for 17.6 Compound Interest. On your own paper complete the
homework assignment.
378
17.7a Logarithms – Convert Between Logs and Exponents
Logarithm:
𝑏 𝑥 = 𝑎 can be written as __________________
Example 1:
Example 2:
Write as a log:
𝑚2 = 25
Q 1:
Write as an exponent:
log 𝑥 64 = 2
Q 2:
379
17.7b Logarithms – Evaluate Logs
To evaluate a log: make the equation ______________________ and convert to an ________________
Example 1:
Example 2:
1
log 3 ( )
81
log 4 64
Q 1:
Q 2:
380
17.7c Logarithms – Solve Log Equations
To solve a log equation: convert to an ________________
Example 1:
Example 2:
log 5 (2𝑥 − 6) = 2
log 𝑥 8 = 3
Q 1:
Q 2:
You have completed the videos for 17.7 Logarithms. On your own paper complete the homework
assignment.
Congratulations! You made it through the material for Unit 17: Functions. It is time to prepare for
your exam. On a separate sheet of paper complete the practice test. Once you have completed the
practice test, ask your instructor to take the test. Good luck!
381
Unit 18:
Proficiency Exam #3
To work through this unit you should:
1. Complete the practice tests on your own paper.
2. Take the (two part) unit exam.
382
Conversion Factors
LENGTH
English
Metric (meter)
12 in = 1 ft
3 ft = 1 yd
1 mi = 5280 ft
1000 mm = 1 m
100 cm = 1 m
10 dm = 1 m
1 dam = 10 m
1 hm = 100 m
1 km = 1000 m
TEMPERATURE
English to Metric
1 in = 2.54 cm
VOLUME
English
Metric (liter)
8 fl oz = 1 cup (c)
2 cups (c) = 1 pint (pt)
2 pints (pt) = 1 quart (qt)
4 quarts (qt) = 1 gallon (gal)
1000 mL = 1 L
100 cL = 1 L
10 dL = 1 L
1 daL = 10 L
1 hL = 100 L
1 kL = 1000 L
1 mL = 1 cc = 1 cm3
TIME
60 seconds (sec) = 1 minute (min)
60 minutes (min) = 1 hour (hr)
24 hours (hr) = 1 day
52 weeks = 1 year
365 days = 1 year
English to Metric
1 gallon (gal) = 3.79 liter (L)
1 in3 = 16.39 mL
WEIGHT (MASS)
English
Metric (gram)
16 oz = 1 pound (lb)
2,000 lb = 1 Ton (T)
1000 mg = 1 g
100 cg = 1 g
10 dg = 1 g
1 dag = 10 g
1 hg = 100 g
1 kg = 1000 g
INTEREST
.
Simple:
Continuous:
A = Pert
Compound:
Annual: n=1
Semiannual: n=2
English to Metric
Quarterly: n=4
2.20 lb = 1 kg
383
.
Geometric Formulas
Name
Rectangle
Diagram
Area
w
A = lw
l
h
Parallelogram
A = bh
b
Triangle
h
b
a
h
Trapezoid
P = 2l + 2w
A=
bh
A=
h(a + b)
b
d
Circle
Name
r
A = πr2
Diagram
C = πd = 2πr
Volume
Rectangular
Solid
h
l
w
V = lwh
r
Right Circular
Cylinder
V = πr2h
h
Right Circular
Cone
h
V=
πr2h
V=
πr3
r
Sphere
r
Right Triangle
Pythagorean Theorem:
a2
+
b2
=
384
c2
c
a
b