Bails – Math 094 Notes
Pre-Algebra Review of Order of Operations on Whole Numbers (Optional)
The Order of Operations is a set of rules to simplify expressions. Without order of operations, we
could have different answers for one expression. Take the example below:
A.
2+4•5
6•5
30
B.
2+4•5
2 + 20
22
Which answer is correct? A or B
Order of Operations:
Parentheses also known as grouping symbols: ( ), [ ], { }.
Exponents (more on this later)
⎧Multiply ⎫
⎨
⎬
⎩Divide ⎭
These two operations should be completed in
order from left to right.
⎧ Add
⎫
⎨
⎬
⎩Subtract ⎭
These two operations should be completed in
order from left to right.
Based on the rules for order of operations, the correct answer to 2 + 4 • 5 is 22.
Use the order of operations to simplify without a calculator.
1.
2.
3 + 2(10 – 7)
For this problem:
= 3 + 2(_______)
Work the inside of the parentheses first.
= 3 + _______
Now multiply
= _______
Add last
25 ÷ 5 • 5
For this problem:
= _______ • 5
Always do the multiplication and division in order from left to
right. Since division comes first in this problem, we need to
divide before we multiply.
= _______
1
Chapter 1 Introduction to Algebra: Integers
Use the order of operations to simplify without a calculator.
3.
24 + 6 ÷ 2
4.
30 − (13 + 2 )
5.
15 − 5 i 3
6.
25 − 3 i 7 + 4
7.
4i7+3i5
8.
36 ÷ 9 + 8 − 5
9.
(14 + 28 ) − ( 34 − 27 )
10.
100 ÷ 4 i 5
11.
17 + 5 ( 6 )
12.
( 56 − 8 ) − (17 + 7 )
2
Bails – Math 094 Notes
Section 1.1 Place Value
Digits: the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 make up all possible numbers.
1.
351 is a ________digit number
1,207 is a ________digit number
2.
Whole numbers are the set of numbers {0, 1, 2, 3, 4, 5 …}. Notice that there are no negative
numbers, fractions or decimals included in this set.
3.
Circle the whole numbers:
100
−10
15
3
4
3.25
0
−2.5
4.
Circle the whole numbers:
−2
3000
1
5
18
0.3
−35
8.25
Place Value Table
Millions
Column
Hundred
Thousands
Column
Ten
Thousands
Column
Thousands
Column
Hundreds
Column
Tens
Column
Ones
Column
1,000,000
100,000
10,000
1,000
100
10
1
5.
6.
For the number 245,689 find the place value for the following digits:
2: Place Value =
4: Place Value =
6: Place Value =
9: Place Value =
For the number 5,762,034 find the place value for the following digits:
0: Place Value =
4: Place Value =
5: Place Value =
7: Place Value =
Write each number in words.
7.
409
8.
8019
3
Chapter 1 Introduction to Algebra: Integers
Write each number using digits.
9.
Seventy-eight thousand, four hundred nine
10.
Six million, five hundred sixty-two thousand, two hundred sixteen
11.
Nine hundred twenty-five thousand, one hundred thirty-two
12.
Three million, five hundred
13.
Twelve million, two
Write each number in words.
14.
10,000,050
15.
9,456,321
16.
701,816
17.
9,059,046
4
Bails – Math 094 Notes
Section 1.2 Introduction to Signed Numbers
Write each of the following as a positive or negative number.
1.
Julie overdrew her checking account by $82.
2.
The temperature dropped fifteen degrees overnight.
3.
Bart gained seven pounds over the holidays.
4.
The Bears lost eight yards on the first play of the game.
Real Number Line
Label the origin
Label the positive numbers
Label the negative numbers
Graph each set of numbers.
5.
−4, 3, 0, − 1
6.
−5, − 2, 4, 5
7.
1 1
−3 , , 0, − 4
2 2
8.
1
−7, − 5, 2 , − 3
2
5
Chapter 1 Introduction to Algebra: Integers
> Greater than symbol
Inequalities
< Less than symbol
The number to the left on the real number line is always the smaller value.
In the space provided, place either < or > to make the statement true.
Hint: Always point the arrow at the smaller number.
9.
4 _____ 7
10.
4 _____ − 7
11.
−4 _____ 7
12.
−4 _____ − 7
13.
3 _____ 0
14.
−3 _____ 0
15.
−10 _____ − 13
16.
−18 _____ − 8
Absolute Value is the distance from zero on the number line. Notation: ⏐x⏐
Example 1.
⏐3⏐ = 3
Example 2.
⏐–3⏐ = 3
Simplify the following absolute values.
17.
⏐5⏐ = ______
18.
⏐–2⏐ = ______
19.
⏐–10⏐ = ______
20.
⏐0⏐ = ______
In the space provided, place either <, >, or = to make the statement true.
21.
⏐–2⏐ ______ ⏐3⏐
22.
–10 ______ ⏐–10⏐
23.
⏐–5⏐ ______ ⏐–6⏐
24.
⏐–2⏐ ______ 4
25.
–5 ______ –6
26.
⏐10⏐ ______ –10
6
Bails – Math 094 Notes
Section 1.3 Adding Integers
Rules for Adding Negative Numbers
1.
If the signs are the ____________________, add the numbers and copy the sign.
2.
If the signs are ________________________, subtract the smaller number from the larger
number and copy the sign of the larger number.
Simplify the following without a calculator.
1.
10 + ( −20 ) =
2.
10 + 20 =
3.
( −10 ) + 20 =
4.
−10 + ( −20 ) =
5.
−3 + ( −4 ) =
6.
3 + ( −4 ) =
7.
−3 + 4 =
8.
4 + −3 =
9.
−7 + 5 =
10.
−13 + −6 =
11.
14 + −8 =
12.
−9 + 2 =
13.
−7 + −11 =
14.
−18 + 21 =
15.
−2 + −3 + −4 =
16.
−2 + 3 + −4 =
17.
2 + −3 + −4 =
18.
2 + 3 + −4 =
19.
15 + −9 + −6 =
20.
10 + −2 + −7 + 5 =
7
Chapter 1 Introduction to Algebra: Integers
Addition Property of Zero (Additive Identity Property): A + 0 = A
The sum of a number and zero is the original number.
21.
0 + 18 =
22.
or
0+A=A
−36 + 0 =
Commutative Property of Addition: A + B = B + A
In addition, the order does not affect the sum.
Rewrite using the Commutative Property of Addition then check to make sure the sum is
unchanged.
23.
47 + 83 =
24.
−13 + −37 =
Associative Property of Addition: (A + B) + C = A + (B + C)
The grouping of three or more terms does not affect the sum.
Rewrite using the Associative Property of Addition then check to make sure the sum is
unchanged.
25.
( − 6 + 8 ) + −8 =
26.
−17 + ( −3 + 15 ) =
Write an addition problem for each situation and find the sum.
27.
Donna lost five pounds in July then seven pounds in August. In September, Donna
gained three pounds. What is the total number of pounds she gained or lost over the
three months?
28.
Mark had $83 in his checking account. He wrote a check for $57 and was charged
$24 for overdrawing his account last month. What is Mark’s balance?
8
Bails – Math 094 Notes
Section 1.4 Subtracting Integers
Opposites: The opposite of x is –x. To find the opposite of a number, change the sign.
1.
The opposite of 3 is ________
2.
The opposite of –2 is ________
3.
The opposite of –5 is ________
4.
The opposite of a positive number is a ____________________ number.
5.
The opposite of a negative number is a ____________________ number.
Property of Opposites
If x is a positive number, then
(two negatives make a positive)
–(–x) = x
Definition of Subtraction: To subtract a number, add its opposite
a – b = a + (-b)
a –(–b) = a + b
All subtraction problems can be written as ___________________ problems.
Steps for Subtracting Negative numbers:
1.
Change all subtraction problems to addition problems.
→ when subtracting a negative number, remember to change to addition.
example: 3 – (–4) = 3 + 4 = 7
2.
Add negative numbers as before.
→ If the signs are the same, __________________ the numbers and copy the sign.
→ If the signs are different, ___________________ the smaller number from the larger
number and copy the sign of the larger number.
Simplify the following without a calculator.
6.
10 − ( −20 ) =
7.
10 − 20 =
8.
−10 − 20 =
9.
−10 − ( −20 ) =
9
Chapter 1 Introduction to Algebra: Integers
Simplify the following without a calculator.
10.
3−4=
11.
−3 − ( −4 ) =
12.
−3 − 4 =
13.
4 − −3 =
14.
−7 − 5 =
15.
−13 − −6 =
16.
14 − ( −8 ) =
17.
−9 − 2 =
18.
−7 − −11 =
19.
18 − 21 =
20.
−2 − 3 − 4 =
21.
−2 + 3 − 4 =
22.
2 − 3 − ( −4 ) =
23.
2−3 − 4=
24.
15 − 9 − 6 =
25.
−10 + 2 − ( −7 ) − 5 =
26.
6 − −4 =
27.
6 − ( −4 ) =
28.
4−6 =
29.
5−9 − 6 =
10
Bails – Math 094 Notes
Section 1.5 Rounding and Estimating
Rounding Numbers
We would round numbers when describing
• Driving Distance
• Population
• Estimating
Steps for Rounding Numbers:
1.
Place an arrow under the digit you wish to round
2.
Look at the number to the right of the arrow
3.
If the next digit is a ________________ keep the number above the arrow the same
and replace remaining number with zeros.
4.
If the next digit is a ________________ add one to the number above the arrow and
replace remaining numbers with zeros.
Example:
Round 567,234 to the nearest thousand.
567,234
↑
Digit to the right is a 2 so keep number above the arrow the same
567,234 rounded to the nearest thousands is 567,000
Round the following numbers.
1.
4,586 to the nearest tens ____________________
2.
248 to the nearest hundreds ____________________
3.
10,964 to the nearest thousands ____________________
4.
42,318 to the nearest ten thousand ____________________
5.
499,651 to the nearest ten thousand ____________________
6.
39,837,241 to the nearest hundred thousand ____________________
7.
39,837,241 to the nearest million ____________________
8.
39,837,241 to the nearest ten-million ____________________
11
Chapter 1 Introduction to Algebra: Integers
Estimating is used to find an answer quickly. One common method is to round each number so
that you can do the arithmetic mentally.
Estimate the following problems by rounding to the indicated place value.
9.
399 + 610 + 744 + 298 (hundred)
10.
12,455 + 68,321 + 8,477 + 91,670 (ten thousand)
Front End Rounding: In front end rounding, you round the first digit of the number. In other
words, all digits will be 0 except the first digit.
Use front end rounding to estimate each answer. Then find the exact answer.
11.
283 + (–791)
Estimate: __________+ __________ = ____________
Exact Answer = ____________
12.
–3069 + –7122
Estimate: __________+ __________ = ____________
Exact Answer = ____________
13.
489 – (–612)
Estimate: __________+ __________ = ____________
Exact Answer = ____________
14.
–998 – (–1107)
Estimate: __________+ __________ = ____________
Exact Answer = ____________
12
Bails – Math 094 Notes
Section 1.6 Multiplying Integers
We will use a raised dot or parentheses to express multiplication and not the × symbol.
For example, 3i4 or ( 3 )( 4 ) or 3 ( 4 ) . The phrases we use to indicate multiplication are product,
multiply or times.
Multiplication is a short way to write repeated addition.
10 + 10 + 10 = 3(10) = _________
or
(−10) + (−10) + (−10) = 3(−10) = _________
The product of a positive number and a negative number is a ______________________ number
1.
−3(3) =
2.
−3(2) =
3.
−3(1) =
4.
−3(0) =
The product of a negative number and a negative number is a _____________________ number.
5.
−3(−1) =
6.
−3(−2) =
7.
−3(−3) =
8.
−3(−4) =
Rules for multiplying integers
1.
The product of an _______________ number of negatives is _______________.
2.
The product of an _______________ number of negatives is _______________.
Simplify the following without a calculator.
9.
−1(2)( −3) =
10.
−1(−2)( −3) =
11.
(−3)(−1)(−2)(2) =
12.
(−3)(−1)(−2)(−2) =
13.
2(−1)(−5)( −2)( −1) =
14.
(−15)(0)(−7)(−4) =
Multiplication Property of Zero: A • 0 =0
A number multiplied by zero is always zero.
0•A=0
Multiplication Property of One: A • 1 = A
1•A=A
A number multiplied by one is always the original number.
15.
25 i 0 =
16.
25 i 1 =
13
Chapter 1 Introduction to Algebra: Integers
Commutative Property of Multiplication: A • B = B • A
In multiplication, the order does not affect the product.
Rewrite the following using the commutative property of multiplication.
17.
2•3=
18.
5 • −4 =
Associative Property of Multiplication: (A • B) • C = A • (B • C)
The grouping of three or more terms does not affect the product.
Rewrite the following using the associative property of multiplication.
19.
(3 • −1) • −9 =
20.
−5 • (2 • −3) =
Distributive Property: A(B + C) = AB + AC
Multiplication distributes over addition.
Rewrite the following using the distributive property and simplify:
21.
2 (3 + 4 )
22.
3 ( −4 + 10 )
Use rounding to estimate each answer. Then use your calculator to find the exact answer.
23.
–87 • –22
Estimate: __________ • __________ = ____________
Exact Answer = ____________
24.
–62 • 88
Estimate: __________ • __________ = ____________
Exact Answer = ____________
25.
–563 • – 511
Estimate: __________ • __________ = ____________
Exact Answer = ____________
14
Bails – Math 094 Notes
Section 1.7 Dividing Integers
Review of adding and subtracting integers: simplify without a calculator.
1.
−13 − 6 =
2.
−13 − ( −6 ) =
3.
−27 + 11 =
4.
−11 − ( −27 )
We will use the division symbol ÷ or the fraction bar to express expression division; for example,
12
12 ÷ 4 or
. The phrase we use to indicate division is quotient.
4
Rules for dividing integers
1.
The quotient of a negative and a positive is _________________.
2.
The quotient of a negative and a negative is __________________.
Simplify the following without a calculator.
5.
−56 ÷ 8 =
6.
−63 ÷ −9 =
7.
−18
=
−6
8.
49
=
−7
Division Properties
A
= 1 A nonzero number divided by itself is always 1.
A
A
1
0
A
A
0
= A A number divided by one is always the original number.
= 0 Zero divide by any nonzero number is 0.
= Undefined Division by zero is undefined.
Use the division properties to simplify the following.
9.
11.
–12 ÷ –12=
−20
1
=
10.
12.
–12 ÷ (0)=
0
−20
=
15
Chapter 1 Introduction to Algebra: Integers
Combining Multiplication and Division of Integers: If there are parentheses, work inside the
parentheses first. Next, work in order from left to right using two numbers at a time.
Simplify the following without a calculator.
−49 ÷ 7 • −7
13.
14.
25 ÷ ( −5i −5 )
15.
84 ÷ −2 ÷ −3
16.
−8i−9 ÷ 36i−4
17.
−6i −3 ÷ 2 i9
18.
42 ÷ ( 7i3 ) ÷ ( −6 − −6 )
Use rounding to estimate each answer. Then use your calculator to find the exact answer.
19.
26,680 ÷ 58
Estimate: __________ ÷ __________ = ____________
Exact Answer = ____________
20.
18,936 ÷ −18
Estimate: __________ ÷ __________ = ____________
Exact Answer = ____________
21.
If 100 guests attend a party and one pie will serve 8 people, how many pies should be
made so that each person gets one slice? How many pieces will be left over?
Estimate: __________ ÷ __________ = ____________
16
Bails – Math 094 Notes
Section 1.8 Exponents and Order of Operations
Exponent: another way to write repeated multiplication.
1.
3.
3 i 3 = 32
2.
2 i 2 i 2 = 23
The exponent is
The exponent is
The base is
The base is
Read as
Read as
Simplified
Simplified
43
The exponent is
4.
4
The exponent is
The base is
The base is
Read as
Read as
Simplified
Simplified
Write in expanded form then multiply.
5.
(–5)2 = (–5)( –5) =
6.
(–3)3 = (–3)(–3)(–3) =
7.
(–2)4 =
8.
(–9)2 =
Simplify the following.
9.
72 =
10.
( −10 )
11.
25 =
12.
15 =
13.
( −1)
14.
( −1)
15.
4 2 ( −1) =
16.
( −2 ) ( 3 )
3
=
9
12
2
=
=
3
2
=
Use your calculator (Base y x Exponent = ) to simplify the following.
17.
84 =
18.
172 =
19.
153 =
20.
( −4 )
21.
( −9 )
22.
−5 i 222 =
4
=
7
=
17
Chapter 1 Introduction to Algebra: Integers
Order of Operations – Set rules used to simplify expressions.
A.
2+4•5
6•5
30
B.
2+4•5
2 + 20
22
Which answer is correct? A or B
Without set rules there would be several different answers to every problem!
Order of Operations:
P__________________ also known as grouping symbols: ( ), [ ], { }, or fraction bar.
E__________________
⎧M _________________ ⎫
⎨
⎬
⎩D _________________ ⎭
These two operations should be completed in
order from left to right
⎧ A _________________ ⎫
⎨
⎬
⎩S _________________ ⎭
These two operations should be completed in
order from left to right
Use the order of operations to simplify without a calculator.
23.
6i7+4i5
24.
( 3 − 7 )( 4 − 11)
25.
−3 i 4 − 8 i ( −9 )
26.
−19 + 2 ( 3 − 7 )
18
Bails – Math 094 Notes
27.
−4 ( − 3 + 5 i 6 )
28.
15 − 6 ( 7 − 9 ) − 52
29.
8 + 54 ÷ ( −3 ) − ( −2 )
30.
−500 ÷ ( 50 ÷ −10 )
31.
2 ( 25 ÷ 5 i 5 − 3 )
32.
2 ( 8 − 11) − 3 ( −3 + 2 )
2
2
19
Chapter 1 Introduction to Algebra: Integers
33.
−54 ÷ 9 − 2 −7 + 3
34.
14 − 5 − 4 i 6 + ( −8 ) ÷ 4 2
35.
36.
20
2
−22 + 5 ( 4 − 6 )
4 − 42 ÷ 8
−10 + 42 − 6
2 + 3 (1 − 4 )
Bails – Math 094 Notes
Section 2.1 Introduction to Variables
Variable: letter that represents part of a rule that varies.
Constant: part of a rule that does not change.
Coefficient: the number in front of the variable.
Expression: combination of variables and constants terms separated by operations.
For the following expressions, identify the variable term, coefficient, and constant term.
1.
3x − 7
variable term =
coefficient =
constant term =
2.
−4 y + 11
variable term =
coefficient =
constant term =
3.
−x − 6
variable term =
coefficient =
constant term =
4.
y
variable term =
coefficient =
constant term =
5.
−30
variable term =
coefficient =
constant term =
Review of Exponents:
Recall 32 = 3 • 3 = 9
so
3x2 = 3x • x and
Rewrite each expression without exponents.
7.
6.
2x 3
8.
x4y 3
9.
x2y3 = x • x •y • y • y
xy 3
9x 2 yz 3
Steps for finding values of polynomials:
1.
Write the original expression
2.
Copy the original again, but replace the _______________ with a set of parenthesis
3.
Place given value of variable inside parenthesis
4.
Simplify
Evaluate each expression when a = −3 , b = −1, and c = 2
3abc
10.
11.
a3
21
Chapter 2 Understanding Variables and Solving Equations
Evaluate each expression when a = −3 , b = −1, and c = 2
12.
−2a 2 b 2
13.
ab + bc
14.
2a − c
b
15.
3c 2 + 5b
−3 − a
16.
The expression for determining the cost per ounce is c ÷ z where c is the total cost
and z is the number of ounces. Evaluate the expression when the total cost of caviar
is $48 for 16 oz.
17.
The expression for determining the perimeter of a rectangle is 2L + 2W, where L is
the length and W is the width. Evaluate the expression when
a.
The length is 15 centimeters (cm) and the width is 11 centimeters.
b.
The length is 20 feet (ft) and the width is 15 feet.
22
Bails – Math 094 Notes
Section 2.2 Simplifying Expressions
Similar (Like) Terms: have the same variable(s) and exponents. The number in front of the
variable (coefficient) can differ.
Like Terms Examples:
3x, 4x
2xy, -5xy
4y2, 10y2
Unlike Terms Examples:
3x, 3y
3x2, 4x
4x, 5xy
Steps for Adding and Subtracting Polynomials:
1.
Rewrite expression without parenthesis
If necessary, make sure to distribute negative through parentheses
2.
Identify like ______________________
3.
Add/subtract the ______________________ (number in front of variable) & copy the variable
**Variables always stay the same in addition and subtraction
4.
Write each answer with the variables in alphabetical order and any constant term last.
Simplify (combine like terms) if possible.
3x − 8x
1.
2.
x+x+x
3.
3y − 7y + y
4.
2x 2 + 5x 2
5.
−8 xy − 9 xy − xy
6.
2x + 3 + 4 x + 1
7.
3x − 5 − 9x + 2
8.
4y − 5 x + 3y − 7x
9.
−3 x 2 + 4 x − x 2 − 11x
10.
6 x 3 + 7 x 2 + 10 x
23
Chapter 2 Understanding Variables and Solving Equations
Review of Associative Property of Multiplication:
(a • b) • c = a • (b • c)
Simplify by using the associative property of multiplication.
11.
8 ( −3 x ) =
12.
(
)
−9 3 x 2 =
Review of Distributive Property: a ( b + c ) = ab + ac
Note: If there is no number (coefficient) in front of the variable, the number is 1. Therefore, x = 1x
Use the distributive property to simplify each expression.
13.
2 ( 6 x + 1) =
14.
3 ( x − 2) =
15.
4 ( 4 x + 1) =
16.
−5 ( 7 x + 3 ) =
18.
−3 ( x − 2 ) + 15
Simplify each expression.
2 ( 6x − 5) − 2
17.
19.
4 + 6 (9x − 2)
20.
6 − 10 ( x − 1)
21.
16 x − 6 ( 9 x − 2 ) − 18
22.
3 + 5 ( 2 x − 7 ) − 15 x
23.
2 ( 3 x ) − 7 + 4 ( 8 x − 1)
24.
−5 x + 3 ( 2 x − 9 ) − 4 ( 3 x − 2 ) + 11
24
Bails – Math 094 Notes
Section 2.3 Solving Equations Using Addition
Addition Property of Equality
If
A=B
Then
A+C=B+C
A, B, C are algebraic expressions
Adding/subtracting the same number to both sides of an equation keeps the equation balanced.
Note: A solution for an equation is a number that makes the equation a true statement.
Solve each equation and check the answer.
1.
x + 7 = 11
2.
x − 7 = 11
3.
x − 7 = −11
4.
x + 7 = −11
5.
33 = −27 + x
6.
52 = x + 61
7.
x − 17 = 21 − 28
8.
−16 = x − 30 + 14
25
Chapter 2 Understanding Variables and Solving Equations
For each equation, simplify each side (if possible) then solve and check the answer.
9.
x − 8 + 4 = −11 − 9
10.
6x + 4 − 5x = 3
11.
7 x − 8 − 6 x = −3 + 1
12.
3 − 6 − 9 = 9x + 5 − 8x
13.
−6 x + 7 x − −9 = −19 − 8
14.
− x + 3 − 7 x + 9 x = −4 + 7 − 9
26
Bails – Math 094 Notes
Section 2.4 Solving Equations Using Division
Division Property of Equality
If A = B, then
A B
=
C C
as long as C ≠ 0
Dividing the same nonzero number on both sides of an equation keeps the equation balanced.
Solve each equation and check the answer.
1.
6 x = 42
2.
−5 x = 40
3.
−8 x = −48
4.
121 = −11x
5.
−27 x = 0
6.
−14 = − x
For each equation, simplify each side (if possible) then solve and check the answer.
7.
2x + 3 x = 17 + 23
8.
8 x − 15 x = 71 − 15
27
Chapter 2 Understanding Variables and Solving Equations
For each equation, simplify each side (if possible) then solve and check the answer.
9.
−8 ( 2 x ) = 32
10.
−60 = −2 ( −15 x )
11.
−14 x + 10 x = 36 + 44
12.
7 x + 5 x = −7 − 4 + 59
13.
82 − 96 = −5 ( −4 x ) − 8 ( 2 x ) = 3 x
14.
−36 + 36 + 41 − 50 = x + 3 ( 2 x ) − 4 ( 4 x )
28
Bails – Math 094 Notes
Section 2.5 Solving Equations with Several Steps
Steps for solving equations with variables on both sides:
1.
Simplify each side of equation by removing parentheses and combining like terms
2.
Use _____________________ property to move all variables (letters) to one side.
3.
Use _____________________ property to move all constant terms (numbers) to the
opposite side of the equation.
4.
Use division property to get a _____________________ (the number in front of the
variable) of 1.
5.
Check the solution by going back to the original equation.
Solve each equation and show all work.
1.
−5 x + 10 = 55
2.
7 x − 4 = 31
3.
−3 x − 6 = −36
4.
42 = 3 ( x + 7 )
5.
3 x = −2 x + 40
6.
11x = 4 x + 8 − 15
29
Chapter 2 Understanding Variables and Solving Equations
Solve each equation and show all work.
7.
5x + 3 = 2x − 3
8.
−2 x + 7 = −4 x + 1
9.
5 x + 10 = 19 + 8 x
10.
15 x + 1 = −4 x + 20
11.
8 ( 2 x + 1) = 6 ( 3 x − 5 )
12.
7 ( x − 8 ) = 2 ( x − 13 )
13.
12 ( x + 2 ) + 5 = 2 x − 1
14.
x + 28 − 10 = 4 ( x + 6 ) − 27
30
Bails – Math 094 Notes
Section 3.1 Problem Solving: Perimeter
Perimeter (P) is the outside measurement of a figure. Units stay the same. ex. ft, yd or m.
The fence around a back yard is an example of perimeter. To find the perimeter of any object:
Add all the lengths of the sides.
Rectangle
Square
P = 2L + 2W
P = 4s
W
s
L
Find the Perimeter of the following figures. Note: Images are not drawn to scale.
1.
2.
5 ft
6 yd
5 ft
11 yd
12 in
3.
4.
3 in
9 cm
16 cm
13 cm
4m
5.
6.
3m
6m
8m
7 km
3 km
7 km
8 km
3 km
10 km
31
Chapter 3 Solving Application Problems
For the following problems, use the appropriate formula to find the missing measurement.
7.
The perimeter of a square is 348 cm. Find the length of each side.
8.
The perimeter of a rectangle is 62 inches. The length is 14 inches. Find the width.
9.
The perimeter of a rectangle is 166 feet. The width is 37 feet. Find the length.
10.
A fence is 40 ft by 50 feet. Find the perimeter.
11.
The perimeter of the following figure is 40 m. Find the missing side.
10 m
4m
5m
4m
3m
5m
3m
32
3m
1m
Bails – Math 094 Notes
Section 3.2 Problem Solving: Area
Area (A) is the cover/surface of a flat object. Answers are in square units. ex. ft 2 , yd 2 , or m2
Sod and carpeting are examples of area.
Rectangle or square
Parallelogram
h
h
b
A = base • height
(or A = length • width)
b
A = base • height
A = bh
Find the area of the following figures. Note: Images are not drawn to scale.
1.
2.
4.5 yd
5 ft
5 ft
3.
6 yd
11 yd
2m
4.
12 in
3 in
2m
5.
6.
6cm
13 mi
10 cm
5 cm
5 mi
33
Chapter 3 Solving Application Problems
For the following problems, use the appropriate formula to find the missing measurement.
7.
Find the area of a parallelogram with base 6 cm and height 8 cm.
8.
The area of a parallelogram is 168 ft 2 . The base is 24 feet. Find the height.
9.
Find the area of a square dog bed with sides measuring 36 inches.
10.
Sue is re-carpeting her bedroom, which measures 20 feet by 14 feet. If carpeting
costs $4 per square foot, what will be the total cost to re-carpet the bedroom?
11.
After painting the kitchen walls, Marti decides to add a decorative border along the
center of the walls. The room measures 4 yards by 6 yards. If the border paper
costs $2 per yard, what is the total cost to add the border to the kitchen?
34
Bails – Math 094 Notes
Section 3.3 Solving Application Problems with One Unknown Quantity
Write and algebraic expressions, using x as the variable.
Addition
The sum of 5 and a number
1.
2.
Three more than a number
3.
A number increased by five
4.
15 plus a number
5.
The total of 8 and a number
6.
A number added to twelve
7.
17 greater than a number
8.
A number exceeded by three
Subtraction
The difference of a number and 2
9.
10.
Four less than a number
11.
A number minus 25
12.
9 reduced by a number
13.
A number subtracted from ten
14.
A number decreased by 16
16.
Six times a number
18.
Triple a number
Division
The quotient of a number and 7
19.
20.
A number divided by 10
Mix of operations
The sum of twice a number and 8
21.
22.
Twice the sum of a number and 8
24.
Three less than twice a number
Multiplication
The product of 6 and a number
15.
17.
23.
Twice a number
Triple a number plus the number
35
Chapter 3 Solving Application Problems
Steps for Solving Word Problems:
1.
Read the Problem.
2.
Assign a variable to the unknown. Let x =
3.
Write an algebraic equation.
4.
Solve the equation.
5.
Answer the question in a complete sentence.
Solve the following word problems. Make sure to write a “Let Statement”, write the
equation, solve the equation, and write your answer using complete sentences.
25.
If two is subtracted from a number, the result is four. What is the number?
Let x = ______________________________________
Equation: _____________________________________
Answer x = _________
Sentence:
26.
If three times a number is increased by four, the result is negative eight. What is
the number?
Let x = ______________________________________
Equation: _____________________________________
Answer x = _________
Sentence:
36
Bails – Math 094 Notes
Solve the following word problems. Make sure to write a “Let Statement”, write the
equation, solve the equation, and write your answer using complete sentences.
27.
Twice the sum of a number and five is four. What is the number?
Let x = ______________________________________
Equation: _____________________________________
Answer x = _________
Sentence:
28.
Four times the sum of twice a number and six is negative eight. What is the
number?
Let x = ______________________________________
Equation: _____________________________________
Answer x = _________
Sentence:
37
Chapter 3 Solving Application Problems
Solve the following word problems. Make sure to write a “Let Statement”, write the
equation, solve the equation, and write your answer using complete sentences.
29.
If the sum of three times a number and two times the same number is increased
by one, the result is sixteen. What is the number?
Let x = ______________________________________
Equation: _____________________________________
Answer x = _________
Sentence:
30.
The product of nine and a number decreased by seven results in the number
increased by forty-one. What is the number?
Let x = ______________________________________
Equation: _____________________________________
Answer x = _________
Sentence:
38
Bails – Math 094 Notes
Solve the following word problems. Make sure to write a “Let Statement”, write the
equation, solve the equation, and write your answer using complete sentences.
31.
92 subtracted from five times Alison’s age is equal to Alison’s age. How old is
Alison?
Let x = ______________________________________
Equation: _____________________________________
Answer x = _________
Sentence:
32.
When four times John’s age is decreased by 96, the result is twice John’s age.
How old is John?
Let x = ______________________________________
Equation: _____________________________________
Answer x = _________
Sentence:
39
Chapter 3 Solving Application Problems
Solve the following word problems. Make sure to write a “Let Statement”, write the
equation, solve the equation, and write your answer using complete sentences.
33.
Shirley ordered four boxes of candles for her restaurant. One candle was put on
each of the 25 tables. If there were 23 candles left, how many candles were
originally in each box?
Let x = ______________________________________
Equation: _____________________________________
Answer x = _________
Sentence:
34.
The 345 running club members each paid the same amount for their yearly
dues. The club earned $1450 helping organize races and spent $1345 on
mailing costs. The club now has $7695 in their bank account. How much did
each member pay in dues?
Let x = ______________________________________
Equation: _____________________________________
Answer x = _________
Sentence:
40
Bails – Math 094 Notes
Section 3.4 Solving Application Problems with Two Unknown Quantities
Steps for Solving Word Problems with Two Unknowns:
1.
Read the Problem.
2.
Assign a variable to the unknown you know the least about. Let x =
3.
When there is more than one unknown, write an expression using the same variable.
4.
Write an algebraic equation.
5.
Solve the equation.
6.
Answer the question in a complete sentence.
Solve the following word problems. Make sure to write a “Let Statement”, write the
equation, solve the equation, and write your answer using complete sentences.
1.
a.
The sum of two numbers is 90. If one number is 6 less than the two times the other
number, what are the two numbers?
Let _____________ = ______________________________________
Let _____________ = ______________________________________
b.
Equation: ______________________________________
c.
Answer :
d.
Answer in a sentence:
41
Chapter 3 Solving Application Problems
Solve the following word problems. Make sure to write a “Let Statement”, write the
equation, solve the equation, and write your answer using complete sentences.
2.
a.
Jason and Barb were candidates for the county school board. Barb won, with 93
more votes than Jason. The total number of votes cast in the election was 587.
How many votes did Barb and Jason each receive?
Let _____________ = ______________________________________
Let _____________ = ______________________________________
b.
Equation: ______________________________________
c.
Answer :
d.
Answer in a sentence:
3.
a.
Nikki is 22 years older than her daughter Kara. The sum of their ages is 46. How
old are Nikki and Kara?
Let _____________ = ______________________________________
Let _____________ = ______________________________________
b.
Equation: ______________________________________
c.
Answer :
d.
Answer in a sentence:
42
Bails – Math 094 Notes
Solve the following word problems. Make sure to write a “Let Statement”, write the
equation, solve the equation, and write your answer using complete sentences.
4.
a.
Kevin earned $50 more than twice what Geoff earned. If the total salary for both
men was $254, how much did Kevin and Geoff each earn?
Let _____________ = ______________________________________
Let _____________ = ______________________________________
b.
Equation: ______________________________________
c.
Answer :
d.
Answer in a sentence:
5.
a.
The attendance at the Saturday night basketball game was three times the
attendance at Sunday’s game. In all 56,000 fans attended the games. How many
fans were at each game?
Let _____________ = ______________________________________
Let _____________ = ______________________________________
b.
Equation: ______________________________________
c.
Answer :
d.
Answer in a sentence:
43
Chapter 3 Solving Application Problems
Solve the following word problems. Make sure to write a “Let Statement”, write the
equation, solve the equation, and write your answer using complete sentences.
6.
a.
The width of rectangle is 3 feet less than the length, and the perimeter is 22 feet.
What is the length and width of the rectangle?
Let _____________ = ______________________________________
Let _____________ = ______________________________________
b.
Equation: ______________________________________
c.
Answer :
d.
Answer in a sentence:
7.
a.
The perimeter of a rectangular room is 86 yards. If the length of the floor is 5 yards
less than three times the width, what is the length and width of the room?
Let _____________ = ______________________________________
Let _____________ = ______________________________________
b.
Equation: ______________________________________
c.
Answer :
d.
Answer in a sentence:
44
Bails – Math 094 Notes
Section 4.1 Introduction to Signed Fractions
Fraction – Any number that can be written in the form
ex.
a
where a and b are integers and b ≠ 0
b
1 3
−5
, , or
2 4
6
Numerator is the ___________ number of a fraction
Denominator is the _______________ number of a fraction
Proper Fraction is when the numerator is less than the denominator.
Improper Fraction is when the numerator is greater than or equal to denominator.
1.
Circle the proper fractions:
4 9 10 8 5 5 6
, ,
,
, , ,
7 8 3 11 4 1 6
2.
Circle the improper fractions:
7 9 1 8 4 15 8
,
, , ,
,
,
7 10 4 7 13 9 1
3.
For the fraction
x
, ______ is the numerator and ______ is the denominator.
10
4.
For the fraction
−3
, ______ is the numerator and ______ is the denominator.
7
Write a fraction based on the information given.
5.
If 3 out of every 7 people who apply to medical school get accepted, what fraction of
the people who apply get accepted? What fraction do not get accepted?
6.
In a recent 10- kilometer race, 757 runners started the race and 599 finished it.
What fraction of the runners finished the race? What fraction did not finish the
race?
45
Chapter 4 Rational Numbers: Positive and Negative Fractions
Graph each fraction on the number line.
−1 3
,
7.
4 4
8.
−3 5
,
8 8
Equivalent Fractions: Fractions that represent the same number. They have the same value
even though they might look different. If a, b, and c are numbers (b ≠ 0 and c ≠ 0), then
ex.
a a•c
=
b b•c
or
a a ÷c
=
b b ÷c
3 3 • 5 15
=
=
4 4 • 5 20
or
22 22 ÷ 2 11
=
=
34 34 ÷ 2 17
Divide the numerator and denominator by 2.
12
9.
10.
14
102
310
Write the fractions with denominator 6.
2
11.
3
12.
60
72
14.
3
4
Write the fractions with denominator 36.
1
13.
3
The number 1 and fractions
a
= a (for any number a)
1
a
= 1 (for any non zero number a)
a
15.
All integers can be written as fractions.
Write 8 as a fraction:
Simplify the following.
18
=
16.
1
17.
18
=
18
18.
−5
=
−5
19.
18
=
−9
20.
−4
=
5
21.
18
=
−9
46
Bails – Math 094 Notes
Section 4.2 Writing Fractions in Lowest Terms
Prime Number: a number that is only divisible by 1 and itself and is greater than 1.
{2, 3, 5, 7, 11, 13, 17, 19, 23, ...}
Composite Number: a number that has a divisor other than 1 and itself and is not a prime
number. All composite numbers can be written as products of prime numbers.
Note: The numbers 0 and 1 are neither prime nor composite.
Steps for Division Method
Divide by prime numbers starting with the
smallest prime number that divides evenly.
Stop when the quotient is 1.
Steps for Factor Tree Method
Write the number as a product of two numbers.
Keep breaking down numbers until each
“branch” ends in a prime number.
Find the prime factorization of each number.
1.
Repeated Division
2.
12
3.
Repeated Division
12
4.
72
5.
Repeated Division
108
Tree Method
Tree Method
72
6.
Tree Method
108
47
Chapter 4 Rational Numbers: Positive and Negative Fractions
Write the prime factorization of each number.
7.
40
9.
Circle the numbers that are prime: 34, 39, 41, 53, 57, 65, 99, 201
10.
Circle the numbers that are composite: 27, 38, 43, 61, 63, 77, 95, 121
8.
48
Reducing Fractions
Method 1 – Write as a product of primes and cancel prime numbers
Method 2 – Divide by common factors
Use either method to reduce the fractions to lowest terms.
11.
12
=
32
12.
36 x 2
=
60 x
13.
15ab 2
=
12ab
14.
36 xy 2
=
20 x 2 y
15.
84 x 2 y 3 z
=
66 xy 3 z 4
48
Bails – Math 094 Notes
Section 4.3 Multiplying and Dividing Signed Fractions
Multiplying Fractions:
a c a•c
• =
for b, d ≠ 0
b d b•d
(multiply top & multiply bottom)
Multiply the following.
1.
5 7
•
6 4
2.
1 2
•
3 7
3.
−4 2
•
5 3
4.
5 4
•
6 5
Steps for multiplying fractions
1.
Look for terms that are divisible by the same number in the numerator and denominator.
2.
Cancel numerator to denominator (divide top and bottom by the same number)
3.
Multiply top, multiply bottom
4.
Make sure answer is in lowest terms.
Find the product and write the answer in lowest terms.
5.
⎛ −2 ⎞ ⎛ −25 ⎞
⎜ 5 ⎟⎜ 4 ⎟
⎝ ⎠⎝
⎠
6.
5 36
•
6 15
7.
60 55
•
22 10
8.
2 13 −10
•
•
13 5
6
9.
1
of (15 x )
5
10.
x y
•
y z
11.
2
3x 2
•
15 x
4
12.
ab 2
c
• 2
c
ab
49
Chapter 4 Rational Numbers: Positive and Negative Fractions
a b
• =1
b a
Two numbers are Reciprocals if their product is 1.
Steps for Dividing Fractions
1.
2.
3.
4.
5.
a c a d
÷ = •
b d b c
Write the original question.
Keep first fraction the same.
Change ÷ to •
“Flip” the second fraction (reciprocal of second fraction).
Multiply as before.
Find the quotient and write the answer in lowest terms.
13.
4 3
÷
3 4
14.
6
÷ ( −2)
5
15.
25 15
÷
24 36
16.
5a 2 b 20a
÷
c
c
17.
12a 2 6a
÷ 2
5b
5b
18.
−5 x 15 x 3
÷
12y 2 −8 y
19.
A certain size bottle holds exactly
4
pint of liquid. How many of these bottles can be
5
filled from a 20-pint container?
20.
How many pieces of pipe that are
feet long?
50
2
ft long must be laid together to make a pipe 16
3
Bails – Math 094 Notes
Section 4.4 Adding and Subtracting Signed Fractions
Adding and subtracting fractions with the same denominators
1.
Copy original
2.
Add/subtract numerator and copy the denominator
3.
Simplify (write answer in lowest terms)
a b a+b
+ =
c c
c
a b a−b
− =
c c
c
Find each sum or difference.
−4
1.
9
3
3.
20
+
−
7
2.
9
1
20
−
4
20
4.
x+5
4
5
x
+
4
x
−
3
−
15
4
x
Warm up to adding/subtracting fractions with unlike denominators. The Least Common Multiple
(LCM) of two numbers is the smallest number (not zero) that is a multiple of both numbers.
List the multiples of the following numbers and circle the first number in common.
5.
6.
4 and 6
6 and 9
7.
4,
6,
6,
9,
12 and 18
8.
12 and 16
12,
12,
18,
16,
Least Common Denominator (LCD) is the smallest number that all denominators can divide into
evenly. This number can be found using the above method.
51
Chapter 4 Rational Numbers: Positive and Negative Fractions
Adding and subtracting fractions with unlike denominators
1.
Find Least Common Denominator.
2.
Rewrite each fraction as equivalent fractions with same denominators.
3.
Add/subtract numerator and copy the denominator
4.
Simplify (write answer in lowest terms)
Find each sum or difference.
9.
11.
13.
15.
17.
52
1
3
5
6
−
x
2
+
3
−
1
4
4
⎛ 7 ⎞
+ ⎜− ⎟
16 ⎝ 12 ⎠
9
−
1
5
10.
12.
14.
16.
−5
5
12
2
+
6
−
9
11
18
3−
5
2
5
3
+
7
x
1
3
1
in on Monday, in on Tuesday, and in on Wednesday. How
2
8
4
much did it grow in the three days?
A corn stalk grew
Bails – Math 094 Notes
Section 4.5 Problem Solving: Mixed Numbers and Estimating
Mixed Number: the sum of a whole number and a fraction
ex.
3+
1
1
=3
4
4
Note: 2
1
⎛ 1⎞
is not the same as 2 ⎜ ⎟
9
⎝9⎠
Steps to change mixed numbers to improper fractions
1.
Multiply whole number and denominator
2.
Take answer from step 1 and add to numerator (this becomes the new numerator)
3.
Keep denominator the same
Change the following to improper fractions.
5
3
1.
8
3.
11
−1
12
6
7
2.
2
4.
3
11
10
Steps to change improper fractions to mixed numbers
1.
Divide denominator into numerator – this becomes the whole number
2.
The remainder in the division becomes the new numerator
3.
Keep the denominator the same
Change the following to mixed numbers.
15
5.
8
6.
26
7
8.
319
20
7.
41
15
9.
1 −9
1
and 1
Graph each fraction on the number line. −2 ,
2 2
3
53
Chapter 4 Rational Numbers: Positive and Negative Fractions
Steps for multiplying/dividing mixed numbers
1.
Change all mixed numbers to improper fractions
2.
Multiply and Divide as before
3.
Make sure answer is in lowest terms
Perform the indicated operations.
10.
5 4
1 •1
6 5
11.
3
1
5
•2
2
6
12.
4
5
1 ÷2
5
6
13.
3
1
2
÷2
2
5
Steps to Adding/Subtracting Mixed Numbers Method 1
1.
Change all mixed numbers to improper fractions
2.
If necessary, find LCD and write as equivalent fractions
3.
Add or Subtract as before
4.
Make sure answer is in lowest terms
Steps to Adding Mixed Numbers Method 2
1.
If necessary, find LCD and write as equivalent fractions. Keep whole numbers the same.
2.
Add whole numbers, add numerators, and keep denominators the same
3.
Make sure answer is in lowest terms
Steps to Subtracting Mixed Numbers Method 2
1.
If necessary, find LCD and write as equivalent fractions. Keep whole numbers the same.
2.
Make sure first numerator is larger than second numerator, if not borrow from the first
whole number to add to the first numerator.
3.
Subtract whole numbers, subtract numerators, and keep denominators the same
4.
Make sure answer is in lowest terms
54
Bails – Math 094 Notes
Perform the indicated operations.
14.
5
2
3
+3
7
7
15.
8
5
5
+9
6
6
16.
5
1
1 +2
8
2
17.
4
1
1
+7
5
3
18.
12
19.
7
1
5
−6
6
6
7
5
−3
8
6
55
Chapter 4 Rational Numbers: Positive and Negative Fractions
Perform the indicated operations.
5
9−6
20.
7
21.
1
2
−1
5
3
3
1
m by 3 m . What is the area of the room?
4
3
22.
A living room has dimensions 3
23.
Erin worked 7 hours on Friday and 5
24.
A skirt requires 3
56
6
3
hours on Saturday. How much longer did
8
Erin work on Friday than on Saturday?
1
yd of material. How much material is needed for 6 skirts?
8
Bails – Math 094 Notes
Section 4.6 Exponents, Order of Operations, and Complex Fractions
Write the following exponents in expanded form then simplify.
2
1.
⎛2⎞
⎜ ⎟
⎝3⎠
3.
⎛ −5 ⎞ 12
⎜ ⎟ •
⎝ 6 ⎠ 15
3
2.
⎛ −1 ⎞
⎜ ⎟
⎝5⎠
4.
⎛3⎞ ⎛4⎞
⎜ ⎟ ⎜ ⎟
⎝8⎠ ⎝3⎠
2
2
2
Order of Operations:
P__________________ also known as grouping symbols: ( ), [ ], { }, or fraction bar.
E__________________
⎧M _________________ ⎫
⎨
⎬
⎩D _________________ ⎭
These two operations should be completed in
order from left to right
⎧ A _________________ ⎫
⎨
⎬
⎩S _________________ ⎭
These two operations should be completed in
order from left to right
Simplify the following.
3 8 1
i −
5.
4 9 3
2
6.
⎛ 1 ⎞ ⎛ −1 ⎞
⎜3⎟ −⎜ 2 ⎟
⎝ ⎠ ⎝ ⎠
3
57
Chapter 4 Rational Numbers: Positive and Negative Fractions
Simplify the following.
7.
−7
16
−
⎛3⎞
−⎜ ⎟
8 ⎝4⎠
1
2
8.
2
2
2
+1 i 3
3
3
5
9.
⎛1 3 ⎞
⎛ 1 1⎞
5⎜ +
− 2⎜
+ ⎟
⎟
⎝ 5 10 ⎠
⎝ 10 2 ⎠
10.
1⎞
⎛ −3 5 ⎞ ⎛ 3
⎜ 4 + 8 ⎟ ⎜ 2 8 − 14 ⎟
⎝
⎠⎝
⎠
58
Bails – Math 094 Notes
Simplify the following.
1⎞
⎛ 1
10 − 5 ⎜ 7 ÷ 2 ⎟
11.
4⎠
⎝ 5
12.
5
1 3 1
i4 ÷ −
8
2 4 3
13.
4
14.
⎛ 3 ⎞ ⎛ 5 2 ⎞ ⎛ −1 ⎞
⎜5⎟ ⎜9 − 3⎟÷⎜ 5 ⎟
⎝ ⎠ ⎝
⎠ ⎝ ⎠
1 3
1 1
÷ −5 i1
2 4
4 7
2
2
59
Chapter 4 Rational Numbers: Positive and Negative Fractions
Complex Fractions are fractions in which the numerator and denominator contain one or more
fractions. Note: all complex fractions can be written as division problems.
Simplify the following by rewriting as division problems
5
15.
18
2
9
16.
−20
2
5
2
17.
⎛2⎞
⎜3⎟
⎝ ⎠
2
⎛ −4 ⎞
⎜ 5 ⎟
⎝
⎠
1
18.
2
3
4
60
+
+
2
3
5
6
Bails – Math 094 Notes
Section 4.7 Problem Solving: Equations Containing Fractions
Addition Property of Equality
If
A=B
Then
A+C=B+C
A, B, C are algebraic expressions
Adding/subtracting the same number to both sides of an equation keeps the equation balanced.
Multiplication Property of Equality
If
A=B
Then
A•C = B•C
A, B, C are algebraic expressions and C ≠ 0
Multiplying/dividing the same number on both sides of an equation keeps the equation balanced.
Steps for solving equations
1.
Simplify each side of equation by removing parentheses and combining like terms
2.
Use addition property to move all variables to one side.
3.
Use addition property to move all constant terms to the opposite side of the equation.
4.
Use multiplication/division property to get a coefficient of 1.
5.
Check the solution by going back to the original equation.
Solve the following equations.
1.
3.
5.
1
2
x = −4
−4
5
x=−
2.
8
15
1
x+7=4
3
4.
6.
2
3
2
3
x = 18
x=
14
12
1
x − 3 = −7
6
61
Chapter 4 Rational Numbers: Positive and Negative Fractions
7.
2
x −1= 7
5
9.
−4 + 7 +
11.
4x +
62
1
x = −11 + 5
3
7 1
=
9 3
5
x+2
2
8.
−8 =
10.
9−5 =
12.
3x −
1
x +3−7
2
2 5
=
3 6
Bails – Math 094 Notes
Section 4.8 Geometry Applications: Area and Volume
Area of a Triangle
A=
1
bh
2
height
base
1.
Find the area of a triangle with base 13 inches and height 10 inches.
2.
Find the area of a triangle with base
3.
Find the perimeter and area of the triangle.
14
15
inches and height
inches.
5
7
10 m
9m
8m
11 m
4.
Find the perimeter and area of the triangle.
10 ft
12
1
ft
2
12
11
1
ft
2
1
ft
2
63
Chapter 4 Rational Numbers: Positive and Negative Fractions
5.
Find the perimeter and area of the triangle.
6
4 cm
5 cm
3
6.
1
cm
2
1
cm
4
Find the perimeter and area of the figure.
6 cm
6
5 cm
2
cm
5
10 cm
Volume (V) – measure of a space. Answers are in cubic units. ex. ft 3 , yd 3 , or m 3
The amount of water in a pool is an example of volume.
h
Volume of a Rectangular Solid
V = base • height • depth = BHD
or V = length • width • height = LWH
d
b
7.
Find the volume of a rectangular solid with dimensions 3 ft by 2
8.
Find the volume of a cube with sides measuring
64
1
miles.
5
1
1
ft by
ft.
2
4
Bails – Math 094 Notes
Section 5.1 Reading and Writing Decimal Numbers
Quick review of order of operations: simplify without a calculator.
1.
7 + 2 ( −6 + 9 )
3.
−36 ÷ 18i2 − 36
2
2.
5 − 6 ( 2 − 7)
4.
4 − 7 − ( 4 − 7)
2
Decimal Numbers are numbers that fall to the right of a whole number.
Ones
Decimal
Point
Tenths
Hundredths
Thousandths
Ten Thousandths
Hundred Thousandths
Identify the digit that has the given place value.
5.
31.769
6.
128.6409
Ones
Tens
Hundredths
Ten-thousandths
Tenths
Hundredths
Give the place value of the 5 in each of the following numbers:
7.
1.5
8.
32.625
9.
509.067
10.
0.12345
Write the decimal number that has the specified place values.
9 ones, 5 hundredths, 3 tens, 0 tenths.
11.
12.
8 tens, 0 tenths, 1 hundredth, 3 hundreds, 6 thousandths, 2 ones.
65
Chapter 5 Rational Numbers: Positive and Negative Decimals
Writing decimals in words:
1.
Write the whole number
2.
Use the word _____________ for the decimal point
3.
Write the number with the correct place value and end word in “ths”
Write the following in words.
13.
6.2
14.
10. 001
15.
37.0005
Write the following as a decimal number.
16.
Fourteen and five hundredths
17.
One thousand one and two thousand fifteen ten-thousandths
18.
One hundred and seven hundred-thousandths
Write the following as a fraction or a mixed number and reduce your answer.
19.
0.02
20.
0.17
21.
14.035
22.
25.0025
66
Bails – Math 094 Notes
Section 5.2 Rounding Decimal Numbers
Quick review of fractions: simplify without a calculator.
3 3
3 3
+
i
2.
1.
4 4
4 4
3.
3 4
+
4 3
4.
3 3
÷
4 4
5.
3
3−
4
6.
1 ⎛3⎞
−4 ÷ ⎜ ⎟
2 ⎝4⎠
7.
1
⎛2 7 ⎞
+ 4⎜ −
⎟
6
⎝ 5 10 ⎠
8.
⎛ −2 ⎞ ⎛ 1 1 ⎞ 2 ⎛ 1 ⎞
⎜ 3 ⎟ ⎜8 − 2⎟ − 3⎜8⎟
⎝ ⎠ ⎝
⎠
⎝ ⎠
2
3
67
Chapter 5 Rational Numbers: Positive and Negative Decimals
Rounding decimal numbers
1.
Place an arrow under place value to be rounded
2.
Look at the number to the right of specified place value
a.
If the number is 0, 1, 2, 3, 4, keep the number and drop all numbers to the
right of the rounded place value.
b.
If the number is 5, 6, 7, 8, 9, add one to the correct place value and drop all
numbers to the right.
9.
Round 15.453 to the nearest hundredth.
10.
Round 98.766 to the nearest tenth.
11.
Round 495.989 to the nearest ten.
12.
Round 1,234.00579 to the nearest ten-thousandths.
13.
Round 1,299,761.982256 to the nearest ten-thousand.
14.
Round 0.5479998 to the nearest hundred-thousandths.
15.
Round 5,355,601.962498 to the nearest hundred-thousand.
16.
Round 204.987 to the nearest one.
17.
Round 0.9873 to the nearest thousandth.
68
Bails – Math 094 Notes
Section 5.3 Adding and Subtracting Signed Decimal Numbers
To add or subtract decimals, line up the decimals and add or subtract as usual
Ex.
3.09 + 2.1 + 10.016
→
3.090
2.100
+ 10.016
15.206
Notice that we can fill in zeros on the right to help keep the numbers in the correct columns. This
does not change the value of any of the numbers.
Find the sum or difference.
1.
13.29
4.01
2.
+7.87
3.
10.2 + 3.7 + 2 + 4.11
+0.987
4.
5.31
5.
7.
7
6.113
15.76 − 4.61
8.002
−2.87
6.
−6.145
8 − 6.2
8.
5.9 − 3.0126
69
Chapter 5 Rational Numbers: Positive and Negative Decimals
Find the sum or difference. Watch your signs!
9.
0.058 – (–3.08)
10.
–12 – 9.77
11.
–9.87 + 12.97
12.
1.04 – 8.97
13.
–13.1 + (–15.397)
14.
–2.459 – (1.278 – 2.168)
15.
A person making $7.29 per hour is given a raise to $8.05 per hour. How much is
the raise?
16.
Brian agreed to work 37.5 hours this week. If Brian has already worked 18.75
hours, how many more hours must he work?
17.
At a recent visit to the veterinarian, Kelly had to pay $33 for the office visit, $76.90
for vaccinations, $28.85 for a heartworm test, and $7.35 for one month of heart
guard treatment for her dog. What was the final bill?
70
Bails – Math 094 Notes
Section 5.4 Multiplying Signed Decimal Numbers
Steps for Multiplying Decimals
1.
Ignore decimals and multiply as before
2.
Count the number of digits after the decimal in the original problem and make sure the
answer has the same number of digits after the decimal.
Find each product.
0.07(0.03)
1.
2.
–0.42(6)
3.
3.12(0.05)
4.
–0.02(–1.39)
5.
0.5(22)
6.
(–0.3)2
7.
(–0.1)4
8.
1.22
9.
10(.314)
10.
100(.314)
71
Chapter 5 Rational Numbers: Positive and Negative Decimals
Use front end rounding to estimate the answer, translate the word problem into an
algebraic expression, find the exact answer, and answer in a sentence.
11.
What is the perimeter of a rectangular field that measures 4.2 miles by 3.1 miles?
Estimate
Expression
Exact Answer
Sentence
12.
What is the area of a rectangular field that measures 4.2 miles by 3.1 miles?
Estimate
Expression
Exact Answer
Sentence
13.
John worked 63.5 hours over the past two weeks. John earns $9.50 per hour. How
much money did John earn?
Estimate
Expression
Exact Answer
Sentence
14.
Jill filled up the gas tank in her new hybrid car. Jill put 10.39 gallons of gas in her
car at a rate of $2.39 per gallon. How much did she pay for gas?
Estimate
Expression
Exact Answer
Sentence
15.
Find the volume of a rectangular cube with dimensions 3.2 ft by 4.1 ft by 1.1 ft.
Estimate
Expression
Exact Answer
72
Bails – Math 094 Notes
Section 5.5 Dividing Signed Decimal Numbers
Long division with whole numbers: find the quotient.
1.
7 1491
2.
5 2033
Steps for dividing by whole numbers
1.
Use long division as if there were no decimal point involved.
2.
If necessary, add zeros after last digit in the decimal.
3.
Place decimal point directly above the decimal point in the problem.
Find the quotient
3.
37.2 ÷ 8
4.
31.48 ÷ 4
Steps for dividing by decimals numbers
1.
Change divisor to a whole number by moving both numbers the same amount of digits
2.
Divide as before
Find the quotient
5.
0.086 ÷ 0.04
6.
0.00125 ÷ −0.5
73
Chapter 5 Rational Numbers: Positive and Negative Decimals
When rounding make sure to calculate one digit past required place value.
Divide and round the answer to the nearest hundredth:
7.
3.2456 ÷ 0.07
Simplify.
8.
4.579 + 34.8 ÷ (–4)
9.
6 – (0.3)2 ÷ 10
Use front end rounding to estimate the answer, translate the word problem into an
algebraic expression, find the exact answer, and answer in a sentence.
10.
Jill filled up the gas tank in her new hybrid car. Jill put 10.3 gallons of gas in her
car and she had driven 473.8 miles. How many miles per gallon did her car get?
Estimate
Expression
Exact Answer
Sentence
11.
Matt delivers newspapers to gas stations. Last week he paid the publisher
$311.60 for 380 copies. What was the cost per copy?
Estimate
Expression
Exact Answer
Sentence
74
Bails – Math 094 Notes
Section 5.6 Fractions and Decimals
Writing Fractions as Decimals
1.
Change fraction to a division problem
2.
Divide as before
Terminating Decimals are decimal numbers that end.
Write each fraction as a decimal.
1.
3
= 8 3.000
8
2.
14
25
Repeating Decimals are decimal numbers that have a repeating pattern and never end.
Write each fraction as a decimal and use the bar above the repeating digit(s).
3.
5
6
4.
10
11
Write each fraction as a decimal and round to the nearest thousandth.
5.
6
7
Change each decimal to a fraction.
0.06
7.
3
7
6.
2
8.
8.15
75
Chapter 5 Rational Numbers: Positive and Negative Decimals
Find the decimal or fraction equivalent for each number. Write fractions in lowest terms.
9.
Fraction = 3
3
5
Decimal =
10.
Fraction = 4
7
8
Decimal =
11.
Decimal = 0.0045
Fraction =
12.
Decimal = 5.625
Fraction =
In the space provided, place either <, >, or = to make the statement true.
13.
0.1 _____ 0.16
15.
.375 _____
3
8
14.
0.625 _____ 0.0625
16.
7
_____ 0.7
9
Write the following in order from smallest to largest.
17.
76
0.04, 0.39, 0.03, 0.0049, 0.005
18.
0.005, 0.5005, 0.5, 0.049,
1
20
Bails – Math 094 Notes
Section 5.7 Problem Solving with Statistics: Mean, Median, and Mode
Mean (Average) is a number that summarizes a group of numbers.
Steps for finding mean:
1.
Sum the numbers
2.
Divide by the amount of numbers in the set
1.
Find the mean of 1, 4, 6, 8, and 11.
2.
Find the mean of 4, 7, 9, 8, 6, 9, and 13.
Weighted Mean: In a list of data some items may appear more than once, these values are
“weighted” by multiplying it by the number of times it occurs.
Steps for finding mean:
1.
Multiply each number by the frequency.
2.
Sum the frequency and the product columns.
3.
Divide by the product by the frequency
3.
The table below shows the number of students per class and the number of class
sections. Find the weighted mean and round to the nearest whole number.
Students
per class
Frequency
25
3
28
8
30
10
31
2
Product
Totals
4.
The table below shows amount of charity contributions and the number of times the
amount was given. Find the weighted mean and round to the nearest hundredth.
Contribution
value
Frequency
$5
10
$10
13
$12
25
$15
18
Product
Totals
77
Chapter 5 Rational Numbers: Positive and Negative Decimals
5.
Calculating Grade Point Average using a weighted mean. Complete the table and
find the Grade Point Average. Round to the nearest thousandth.
Class
Unit
(credit hour)
Grade
Value
Grade Point
Algebra
4
A
4
16
English
3
B
History
4
C
Music
2
D
Total Units (TU) =
Grade Point Average =
Total Grade Points (TGP) =
TGP
TU
Median (Middle) is the middle number of a set that is written in order from smallest to largest.
Steps for finding median:
1.
Write the numbers in order smallest to largest.
2.
If Odd #, median is middle number.
If Even #, median is mean of middle two numbers
6.
Use the data
41, 48, 52, 61, and 47
to find the following.
Mean
Median
7.
Use the data
30, 42, 36, 50, 22, and 38
to find the following.
Mean
Median
Mode is the number that occurs most often.
8.
Find the mode of 4, 6, 9, 6, 4, 4, and 2
9.
Find the mode of 2, 2, 3, 4, 6, 7, 5, 4, 6, 5, and 2.
78
Bails – Math 094 Notes
Section 5.8 Geometry Applications: Pythagorean Theorem and Square Roots
The square root of a positive number a, written
If
a , is the number we square to get a.
For example 52 = 25 so
a = b then b2 = a
25 = 5
Find each square root without using a calculator.
1.
9
2.
16
3.
36
4.
49
5.
81
6.
144
7.
1
8.
0
Use a calculator to find each square root. Round to the nearest thousandth.
9.
2
10.
5
11.
10
12.
90
13.
325
14.
800
Pythagorean Theorem
Right Triangle – triangle that has a 90° angle
Hypotenuse – longest side of a right triangle
Pythagorean Theorem – the sum of the squares of the two shorter sides of a right triangle is
equal to the square of the longest side.
leg2 + leg2 = hypotenuse2
a 2 + b2 = c 2
or
c = hypotenuse
a = leg
To find the hypotenuse: c = a + b
2
or
hyp =
To find a leg: a = c 2 − b2
or
2
( leg )
2
+ ( leg )
leg =
2
( hyp )
b = leg
2
− ( leg )
2
79
Chapter 5 Rational Numbers: Positive and Negative Decimals
Use your calculator to find the length of the missing side of the following right triangles.
If necessary, round the answer to the nearest tenth.
15.
16.
25 cm
5 yd
7 cm
5 yd
17.
9 mi
18.
6m
11 mi
14 m
19.
Two children are trying to cross a stream. They want to use a log that goes from one
bank to the other. If the left bank is 5 feet higher than the right bank and the stream
is 12 feet wide, how long must a log be to just barely reach?
20.
A wire from the top of a 24-foot pole is fastened to the ground by a stake that is 10
feet from the bottom of the pole. How long is the wire?
80
Bails – Math 094 Notes
Section 5.9 Problem Solving: Equations Containing Decimals
Solve each equation and show your work.
1.
x – 23.2 = –4.5
2.
–7x = –0.63
3.
2x + 3.8 = –7.7
4.
2x – 3.3 = 7x – 5.2
5.
0.7x – 0.32 = 0.5x + 0.56
6.
0.1x + 0.5(x + 8) = 7
81
Chapter 5 Rational Numbers: Positive and Negative Decimals
Solve each equation and show your work.
7.
0.6 x + 4.98 = x − 6.78
8.
3.7 x + 7 = 1.3 x + 13
Solve the following word problems. Make sure to write a “Let Statement”, write the
equation, solve the equation, and write your answer using complete sentences.
9.
Most adult medication doses are for a person weighing 150 pounds. For a 45pound child, the adult dose should be multiplied by 0.3. If the child’s dose of a
decongestant is 16.5 milligrams (mg), what is the adult dose?
Let x = ______________________________________
Equation: _____________________________________
Answer x = _________
Sentence:
82
Bails – Math 094 Notes
Solve the following word problems. Make sure to write a “Let Statement”, write the
equation, solve the equation, and write your answer using complete sentences.
10.
A car rental company charges $12 a day and 18 cents per mile to rent their cars.
If the total charge for a 1-day rental was $33.78, how many miles was the car
driven?
Let x = ______________________________________
Equation: _____________________________________
Answer x = _________
Sentence:
11.
A car rental company charges $11 a day and 18 cents per mile to rent their cars.
If the total charge for a 2-day rental was $61.60, how many miles was the car
driven?
Let x = ______________________________________
Equation: _____________________________________
Answer x = _________
Sentence:
83
Chapter 5 Rational Numbers: Positive and Negative Decimals
Solve the following word problems. Make sure to write a “Let Statement”, write the
equation, solve the equation, and write your answer using complete sentences.
12.
An air compressor can be rented for $38.95 for the first three hours and $8.50 for
each additional hour. If Joe was charged a total of $64.45, how many hours did
he rent the compressor?
Let x = ______________________________________
Equation: _____________________________________
Answer x = _________
Sentence:
13.
The cost for a long distance call is 23 cents for the first minute and 14 cents for
each additional minute. If the total charge for the call is $3.73, how many minutes
was the call?
Let x = ______________________________________
Equation: _____________________________________
Answer x = _________
Sentence:
84
Bails – Math 094 Notes
Section 5.10 Geometry Applications: Circles and Cylinders (not Surface Area)
CIRCLES
Radius (r) – distance from the center to the edge of circle
Diameter (d) – distance from edge to edge through the center
d = 2r
r=
or
d
2
Circumference (perimeter) – distance around the edge of the circle:
Area of a Circle:
π = pi ≈ 3.14
C = 2πr
A = πr2
Volume of a Cylinder:
V = πr2h
h
1.
Find the diameter of a circle with radius 7 inches.
2.
Find the radius of a circle with diameter 22 centimeters.
Use your calculator on the following problems and round to the nearest hundredth.
3.
A circle has a radius of 10 feet.
Find the circumference.
4.
A circle has a radius of 10 feet.
Find the area.
5.
Find the circumference.
6.
Find the area.
8.
Find the area.
3 cm
7.
Find the circumference.
10 in
85
Chapter 5 Rational Numbers: Positive and Negative Decimals
Find the volume.
9.
Find the volume.
10.
2 ft
5m
11 m
10 ft
1.5 ft
11.
Find the volume of a cylinder with radius 5 m, and height 7 m.
12.
Find the volume of a cylinder with diameter 1 in, and height 14 in.
13.
Find the area of the semicircle.
18 cm
14.
An earthquake was felt by people 1000 miles away from the epicenter. How
much area was affected by the quake?
15.
Wayne wanted to buy a pair of walkie-talkies. One model had a range of 3 miles
and the other (much cheaper) model had a range of 2 miles. What is the
difference in the area covered by the 3-mile and 2-mile models?
86
Bails – Math 094 Notes
Section 6.1 Ratios
Ratio is a comparison of the same type of measurements.
Write answers as proper or improper fractions (no decimals or mixed numbers).
a
If a and b are any two numbers, then the ratio of a to b is
(or a : b )
b
Write the following ratios as fractions in lowest terms.
1.
12 to 64
3.
5
5.
1
4
3
to 1
4
2.
3
4
to 0.75
7.
0.8 meters to 0.6 meters
9.
2
1
2
yd to 5 yd
4.
6.
9
5
to
11
5
0.8 to 6.4
3
5
to
7
10
8.
$125 to $2000
10.
1
7 lb to 1 lb
4
87
Chapter 6 Ratio, Rates, and Proportions
Use the chart to find the following ratios.
One cup of breakfast cereal was found to contain the
following nutrients in grams.
11.
Protein to water.
12.
Vitamins to minerals.
13.
Carbohydrates to protein.
14.
Protein to vitamins and minerals.
Find the ratio of the length of the longest to the length of the shortest side.
15.
16.
6 cm
13 in
9 cm
10 in
8 cm
10.5 cm
17.
88
16 in
The price of a frozen pizza increased from $5.50 to $7.00. What is the ratio of the
increase in price to the original price?
Bails – Math 094 Notes
Section 6.2 Rates
Ratio is the comparison of like measurements.
Rate is the comparison of unlike measurements. Write answers as proper/improper fractions (no
mixed number), whole numbers or decimals and include appropriate units in the answer.
Words often used to represent rates:
in
for
on
per
from
Write each rate as a fraction in lowest terms.
1.
276 miles in 6 hours
2.
$110 for 6 visits
3.
210 miles on 12 gallons
4.
166 students in 6 classes
Unit Rate is the rate of two unlike measurements with the denominator = 1. Answers will be
whole numbers or decimals with appropriate units in the answer and a denominator of one.
Find each unit rate.
A train travels 360 miles in 5 hours. Find the rate in miles per hour.
5.
6.
A 50-gallon drum is filled in 46 minutes. Find the rate in gallons per minute.
7.
The flow of water from a facet can fill a 3-gallon container in 15 seconds. Find the
rate in gallons per second.
8.
In 6 hours an airplane travels 4,200 km. What is the rate in kilometers per hour?
89
Chapter 6 Ratio, Rates, and Proportions
Find each unit rate.
1
gallons of gas. Find the rate in mi per gal.
2
9.
A hybrid car travels 675.4 miles on 12
10.
At the beginning of a trip the odometer of a car read 32,567.1 miles. At the end of
the trip it read 32,741.8 miles. If the trip took 4.25 hours, find the rate of the car in
miles per hour.
11.
A 2-liter bottle of root beer costs $1.25. Find the unit price in cents per liter.
12.
An 8-pound bag of dog food costs $10.12. A 25-pound bag of dog food costs
$32.50. Find the unit price in dollars per pound for each bag rounded to the nearest
thousandth. Which bag of dog food is the better buy?
Unit price for small bag of dog food:
Unit price for large bag of dog food:
13.
The 64 ounce container of Ultra Clean laundry detergent is on sale for $3.29. The 48
ounce container of Super Suds is $3.25, but you have a coupon for $0.75 off. Find
the unit price for each laundry detergent rounded to the nearest thousandth. Which
detergent is the best buy?
Unit price for Ultra Clean:
Unit price for Super Suds:
90
Bails – Math 094 Notes
Section 6.3 Proportions
a c
=
b d
Proportion – a statement that two ratios (or rates) are alike.
Solve proportions by cross-multiplying
ad = bc
Determine whether each proportion is true or false by writing the results in lowest terms.
Show the simplified ratios and then circle true or false.
10 55
21 26
=
=
1.
2.
16 88
49 91
True or False
True or False
Use cross products to determine whether each proportion is true or false. Show the cross
products and then circle true or false.
5
20
0.4 1
=
=
3.
4.
0.5 0.2
1.2 3
True or False
True or False
Solve each proportion to find the unknown number.
5.
3 9
=
8 x
6.
30
x
=
300 10
7.
37
x
=
7 14
8.
x 0.4
=
3
8
91
Chapter 6 Ratio, Rates, and Proportions
Solve each proportion to find the unknown number.
9.
2
3 =
y
1
3
5
10.
x
=
12
1
4
1
2
Solve each proportion to find the unknown number and round to the nearest hundredth.
x 105
0.5 1.4
=
=
11.
12.
14 138
0.75
x
Solve each proportion to find the unknown number and write the answer as a whole
number or a fraction. Do not round.
2
9 = x
7
7
3
10
2
13.
92
14.
7
5
1
12 = 8
5
x
6
Bails – Math 094 Notes
Section 6.4 Problem Solving with Proportions
Steps for solving application problems using proportions
1.
Identify unknown (let statement)
2.
Write given info as a ratio and include units.
3.
Write a second ratio so that the units match (top to top and bottom to bottom)
4.
Solve the equation.
5.
Answer the question in a sentence.
Solve the following word problems. Make sure to write a “Let Statement”, write the
proportion, solve the proportion, and write your answer using complete sentences.
1.
Seventy-five magazines cost $65. Find the cost of 12 magazines.
Let x = ______________________________________
Proportion:
Answer x = _________
Sentence:
2.
Craig earns $1557.44 in 16 days. How much does he earn in 246 days?
Let x = ______________________________________
Proportion:
Answer x = _________
Sentence:
93
Chapter 6 Ratio, Rates, and Proportions
Solve the following word problems. Make sure to write a “Let Statement”, write the
proportion, solve the proportion, and write your answer using complete sentences.
3.
A survey showed that 7 out of 8 people want to lose weight. In a group of 752
people, how many would want to lose weight?
Let x = ______________________________________
Proportion:
Answer x = _________
Sentence:
4.
If five pounds of grass seed cover 575 square feet of ground, how many pounds are
needed for 5175 square feet?
Let x = ______________________________________
Proportion:
Answer x = _________
Sentence:
5.
One hundred fifty grams of ice cream contains 13 grams of fat. How much fat is in
325 grams of ice cream, rounded to the nearest whole number?
Let x = ______________________________________
Proportion:
Answer x = _________
Sentence:
94
Bails – Math 094 Notes
Section 7.1 The Basics of Percent
Quick review of decimals/fractions: Without a calculator, write each fraction as a decimal.
If necessary, round to the nearest thousandth.
1.
5
8
2.
3
40
3.
5
7
4.
5
11
Quick review of decimals/fractions: Without a calculator, write each decimal as a fraction.
5.
0.03
6.
2.35
7.
10.007
8.
0.0113
Percent means “per hundred,” so 85% is the same as
85
100
To change a percent to a decimal or a fraction: Drop % symbol and divide by 100.
Change the following percents to decimals.
9.
7%
10.
0.6%
11.
6.24%
12.
1
%
8
13.
99%
14.
1
86 %
5
95
Chapter 7 Percent
Change the following percents to fractions.
15.
40%
16.
2%
17.
34.2%
18.
1
45 %
8
19.
50
%
67
20.
0.45%
To change a decimal or a fraction to a percent: Multiply by 100 and add % symbol.
Change the following numbers to percents.
21.
0.54
22.
2.34
23.
1
4
24.
2
3
25.
1
26.
0.005
27.
3
4
An object weighed on the moon will be only
1
as heavy as it is on earth. Write this
6
number as a percent
28.
In Belgium, 96% of all children between 3 and 6 years of age go to school. In
Sweden, the same figure is only 25%. In the United States, the figure is 60%. Write
each of these percents as a decimal and a fraction.
Belgium
Sweden
United States
96
Bails – Math 094 Notes
Section 7.2 The Percent Proportion
We can solve percent problems using the proportion method.
Percent Proportion Equation
P
A
=
100 B
Example:
or
P
" is "
=
100 " of "
or
P
part
=
100 whole
34 is 25% of what number?
P = 25
A = “is” = 34
B = unknown
25 34
=
100 B
25B = 3400
B = 136
Write a proportion, solve the proportion, and round to the nearest hundredth if necessary.
1.
What is 12% of 80 students?
2.
What percent of 80 cm is 20 cm?
3.
$16 is 20% of what amount?
4.
What is 15% of 110 trees?
97
Chapter 7 Percent
Write a proportion, solve the proportion, and round to the nearest hundredth if necessary.
5.
10% of $22 is how much?
6.
$26 is what percent of $78?
7.
6 pages is 3% of how many pages?
8.
3.75% of 4000 is what number?
9.
249 is 16.9% of what number?
10.
What percent of 123 is 119?
98
Bails – Math 094 Notes
Section 7.3 The Percent Equation (Optional)
Estimating Answers to Percent Problems
• Estimating 200% of a number: Multiply by 2.
• Estimating 50% of a number: Round to the nearest whole number and divide by 2.
• Estimating 25% of a number: Round to the nearest whole number and divide by 4.
• Estimating 10% of a number: Move decimal one place to the left.
• Estimating 1% of a number: Move decimal two places to the left.
Estimate the following answers using the above shortcuts.
1.
200% of 125.
2.
50% of 317.
3.
25% of 44.
4.
10% of 619.
5.
1% of 22.1
6.
300% of 29.
Percent • Whole = Part
Percent Equation:
or
Percent • “of” = “is”
*** You must change the percent to a decimal before using the equation!
Solve each problem using the percent equation. If necessary, round to the hundredth.
7.
What number is 20% of 120?
8.
What percent of 36 is 9?
99
Chapter 7 Percent
Solve each problem using the percent equation. If necessary, round to the hundredth.
9.
37 is 4 % of what number?
10.
What number is 72% of 200?
11.
28% of what number is 56?
12.
26 is what percent of 104?
13.
8 is 2% of what number?
14.
4.89% of 2000 is what number?
15.
9.45 is 15% of what number?
16.
What percent of 78 is 31.9?
100
Bails – Math 094 Notes
Section 7.4 Problem Solving with Percent
Steps for solving application problems involving percent
1.
Read the problem carefully.
2.
Write an equation using either percent proportion or percent equation.
3.
Solve the equation.
4.
Answer the question in a complete sentence.
Solve each problem using the percent proportion or percent equation. If necessary, round
to the hundredth. Make sure to answer in a complete sentence.
1.
If 48% of the students in a certain college are female and there are 2,400 female
students. What is the total number of students in the college?
2.
On a 160-question test a student answered 140 correctly. What percent of the
problems did the student answer correctly?
3.
A family spends $450 every month on food. If the family’s income each month is
$1,800, what percent of the family’s income is spent on food?
101
Chapter 7 Percent
Solve each problem using the percent proportion or percent equation. If necessary, round
to the hundredth. Make sure to answer in a complete sentence.
4.
How much acetic acid is in a 5-liter container of acetic acid and water that is marked
65% acetic acid? How much is water?
5.
In a shipment of airplane parts, 4% are known to be defective. If 34 parts are found
to be defective, how many parts are in the shipment?
6.
If 45 people enrolled in a psychology course but only 35 completed it, what percent
of the students completed the course?
7.
A married couple makes $43,698 a year. If 23.4% of their income is used for house
payments, how much did they spend on house payments for the year?
102
Bails – Math 094 Notes
Finding the Percent Increase or Decrease
P
difference
=
100
original
Solve each problem using the percent increase or decrease formula. If necessary, round to
the hundredth. Make sure to answer in a complete sentence.
8.
When Sue started her job at a local restaurant she made $6.50 per hour. Two
months later she was making $7.20 per hour. What is the percent of increase of her
raise?
9.
The enrollment in a certain elementary school was 410 in 2008. In 2009 the
enrollment in the same school was 328. What is the percent decrease in enrollment
from 2008 to 2009?
10.
During a sale, the price of an entertainment center was cut from $1250 to $999.
What is the decrease in price?
103
Chapter 7 Percent
Section 7.5 Consumer Applications: Sales Tax, tips, Discounts, and Simple Interest
Sales Tax
Tax Rate
Sales Tax
=
100
Cost of Item
Total Cost = Cost of Item + Sales Tax
Discount
Rate of Discount
Discount
=
100
Original Cost
Sale Price = Original Cost – Discount
Tip
15% or 20% tip Amount of Tip
=
100
Dining Bill
Find the tax rate, amount of tax, and total cost. Round to the nearest cent.
Cost of item
Tax rate
Amount of tax
Total cost
1.
2.
= $285.20
= 7.25%
=
=
Cost of item
Tax rate
Amount of tax
Total cost
= $1,620
=
= $64.80
=
For each restaurant bill, estimate a 20% tip, then calculate the exact 15% and 20% tip.
Round to the nearest cent.
Dining Bill
Estimated 20% tip
Exact 15% tip
Exact 20% tip
3.
4.
$12.57
=
=
=
Dining Bill
Estimated 20% tip
Exact 15% tip
Exact 20% tip
$33.85
=
=
=
Find the tax rate, amount of discount, and sale price. Round to the nearest hundredth.
Original price
Rate of discount
Amount of discount
Sale price
5.
= $449.99
= 35%
=
=
6.
104
Original price
Rate of discount
Amount of discount
Sale price
= $85.20
=
= $34.08
=
Bails – Math 094 Notes
Solve each problem using the percent proportion or percent equation and round to the
hundredth if necessary. Make sure to answer in a complete sentence.
Note: These problems may require an additional step.
The purchase price of a cheese and vegetable tray is $24.00 and the sales tax is
7.
$1.56. What is the sales tax rate?
8.
A computer programmer’s yearly income of $68,000 is increased by 8%. What is
her new salary?
9.
A certain brand of peanut butter advertises 15% fewer calories than regular peanut
butter. If the regular peanut butter has 220 calories per two tablespoon serving,
how many calories are in the same-sized serving of light peanut butter?
Simple Interest Formula
i = prt
i = rate of interest, p = principal (amount borrowed or invested), r = rate of percent as a decimal
(ex. 12% = 0.12), and t = time in years
Solve using the simple interest formula.
Sue took out a loan for $2475 at a rate of 8% simple interest for two years. How
10.
much money will she have to pay in interest?
105
Chapter 7 Percent
Solve each problem using the simple interest formula.
1
11.
A local bank pays 3 % simple interest on all savings accounts. If $1800 is
4
invested in this account, how much will be in the account at the end of two years?
12.
Linda needed a short term loan so she borrowed $1,800 from an online bank at a
12% simple interest rate. Linda paid the loan back in 4 months. What was the total
amount of the loan?
13.
Leonard buys a new car for $12,900 (including tax and title) and obtains a loan
through the car dealership. The dealership charges him a simple interest rate of
8.75% for 5 years. What is the total cost of the loan? What are Leonard’s monthly
payments?
Show all work in solving the following problem. Multiple steps are required.
A bedroom set that normally sells for $1,950 is on sale for 18% off. If the sales tax
14.
rate is 7%, what is the total price of the bedroom set if it is bought while on sale?
106
Bails – Math 094 Notes
Section 8.1 Problem Solving with English Measurements
U.S. System (English) Conversions
Convert 10 feet to inches. Since we know 1 foot = 12 inches, a common sense approach would
be to multiply 10 by 12 inches. So 10 ft = 10(12 inches) = 120 in.
These conversions will become very complicated, so we need another way to show conversions
so that we can be certain to end with the correct unit of measure.
1 ft = 12 in written as a Conversion Factor is
1 ft
12 in
or
12 in
1 ft
⎛ 12 in ⎞
10 ft ⎜
⎟ =120 in
⎝ 1 ft ⎠
Convert 10 ft to inches using conversion factors:
Length
Weight
12 inches (in) = 1 foot (ft)
3 feet = 1 yard (yd)
5280 feet = 1 mile (mi)
16 ounces (oz) = 1 pound (lb)
2000 pounds = 1 ton (T)
Liquid Volume
Time
3 teaspoons (tsp) = 1 tablespoon (tbs)
8 fluid ounces (oz) = 1 cup
2 cups (c) = 1 pint (pt)
2 pints = 1 quart (qt)
4 quarts = 1 gallon (gal)
60 seconds (sec) = 1 minute (min)
60 minutes = 1 hour (hr)
24 hours = 1 day
7 days = 1 week
52 weeks = 1 year
365 days = 1 year
Steps for changing units of measure using conversion factors:
1.
Identify starting and ending units
2.
Find the conversion factors that will take you from start to end
3.
Set up multiplication so that all units (except ending) divide out
Make the following conversions in the U.S. system (English system) by multiplying by the
appropriate conversion factors.
1.
28 ft = ____________ in
2.
9.5 yd = ____________ in
107
Chapter 8 Measurement
Make the following conversions in the U.S. system (English system) by multiplying by the
appropriate conversion factors.
3.
132 ft = ____________ yd
4.
126,720 in = ____________ mi
5.
7.2 gal = ____________ pts
6.
1,800 qt = ____________ gal
7.
176 oz = ___________ lbs
8.
1
9.
A 20 ounce package of chocolate chips is on sale for $4.89. What is the cost per
pound, to the nearest cent?
10.
A family with a swimming pool puts up a chain-link fence around the pool. The fence
forms a rectangle 12 yards wide and 24 yards long. If the chain-link fence sells for
$2.50 per foot, how much will it cost to fence all four sides of the pool?
108
3
tons = ___________ oz
4
Bails – Math 094 Notes
Section 8.2 The Metric System – Length
Quick review of application problems: Solve the following word problems. Make sure to
write a “Let Statement”, write the equation, solve the equation, and write your answer
using complete sentences.
1.
If a number is subtracted from five times the number, the result is –16. What is the
number?
Let x = ______________________________________
Equation: _____________________________________
Answer x = _________
Sentence:
2.
While grocery shopping, Rayshawn spent $9 less than four times what Terrance
spent. If Rayshawn spent $23, how much did Terrance spend?
Let x = ______________________________________
Equation: _____________________________________
Answer x = _________
Sentence:
109
Chapter 8 Measurement
Similar units between metric and English system of measurements:
• 1 mile is similar to 1 kilometer (km)
• 1 yard is similar to 1 meter (m)
• 1 inch is similar to 2.5 centimeters (cm)
Choose from km, m, cm, mm.
3.
The distance from home to work is 20 _________.
4.
My wedding ring is 4 _________ wide.
5.
The newborn baby is 50 _________ long.
6.
An aspirin tablet is 10 _________ across.
7.
A paper clip is about 3 _________ long.
8.
The door is 2 _________ high.
Metric System
Kilo
Hecto
Deka
Base Unit
Deci
Centi
Milli
km
kg
kL
hm
hg
hL
dam
dag
daL
Length Meter (m)
Weight Gram (g)
Volume Liter (L)
dm
dg
dL
cm
cg
cL
mm
mg
mL
Helpful acronym to remember the Metric System
K________
H________
D_________ B_________ D________ C_________ M_________
Steps for converting in the metric system
1.
Start in the beginning unit column and move left or right to get to the ending unit.
2.
Count the number of jumps.
3.
Move the decimal the same direction and number as obtained from steps 1 & 2.
Convert the following units in the metric system.
9.
18 m = _____________ mm
10.
4380 m = ____________ km
11.
89.5 cm = ____________ mm
12.
0.23 km = ____________ m
13.
2.1 hm = ____________ cm
14.
349 dm = ____________ m
110
Bails – Math 094 Notes
Section 8.3 The Metric System – Capacity and Weight (Mass)
Quick review of application problems: Solve the following word problems. Make sure to
write a “Let Statement”, write the equation, solve the equation, and write your answer
using complete sentences.
1.
a.
A string is 89 cm long. Marcie’s cat bit the string into two pieces so that one piece is
17 cm longer than the other. What is the length of each piece of string?
Let _____________ = ______________________________________
Let _____________ = ______________________________________
b.
Equation: ______________________________________
c.
Answer :
d.
Answer in a sentence:
2.
a.
A rectangular garden is three times as long as it is wide. The perimeter of the
garden is 96 yards. What is the length and the width of the garden?
Let _____________ = ______________________________________
Let _____________ = ______________________________________
b.
Equation: ______________________________________
c.
Answer :
d.
Answer in a sentence:
111
Chapter 8 Measurement
Similar units between metric and English system of measurements:
• 2 pounds is similar to 1 kilogram
• 1 quart is similar to 1 liter
• 1 mg is a very small unit of measure.
Choose from kg, g, and mg.
3.
Ray’s suitcase weighed 20 _________.
4.
Taylor took a 350 _________ aspirin tablet.
5.
Jenny mailed a letter that weighed 30 _________.
6.
Larry’s basketball weighed 600 _________.
7.
On his diet, Greg can eat 90 _________ of meat for lunch.
8.
One strand of hair weighs 2 _________.
Choose from L, mL
9.
The glass held 250 _________ of water.
10.
Keena can make 5 _________ of soup in that pot.
11.
Cindy donated 500 _________ of blood today.
12.
Rodney bought the large 2 _________ bottle of coke.
13.
Sidney poured 10 _________ of vanilla extract into a bowl.
14.
Chad took 15 _________ of cough syrup every four hours.
Choose from km, m, cm, mm, L, mL, kg, g, mg.
15.
Collin weighs 75 _________.
16.
I hiked 5 _________ this morning.
17.
This apple weighs 180 _________.
18.
I bought 10 _________ of soda for the picnic.
19.
The bracelet is 16 _________ long.
Convert the following units in the metric system:
20.
9.6 L = ____________ mL
21.
5,000,000 mL = ________ kL
22.
5 kg = ____________ g
23.
979 cg = ______________ g
24.
2700 mL = ____________ L
25.
3 g = ______________ mg
112
Bails – Math 094 Notes
Section 8.4 Problem Solving with Metric Measurements
Perform the indicated operations. Make sure the units are the same.
1.
5 m + 35 cm
2.
10 m 17 cm + 5 m 93 cm
3.
2.5 km – 600 m
4.
2 kg – 45 g
Solving a Metric Application
1.
Read the problem.
2.
Work out a plan
3.
Estimate a reasonable answer.
4.
Solve the problem using metric conversions.
5.
Answer the question in a sentence.
6.
Check the answer.
Solve each application problem. If necessary, round the answer to the nearest hundredth.
5.
Bananas are on sale for $1.19 per kilogram. If Shirley purchased 750 g of bananas,
how much will she pay for the bananas?
113
Chapter 8 Measurement
Solve each application problem. If necessary, round the answer to the nearest hundredth.
6.
A Great Dane is approximately 1.3 m tall; whereas, a Yorkshire terrier is
approximately 40 cm tall. What is the difference in height in centimeters?
7.
Clay has two pieces of rope. One measures 3 m 38 cm and the other measures 4 m
72 cm. How many meters of rope does he have in all?
8.
Floor tiles weigh 225 g. Tim needs 48 tiles to cover the bathroom floor. How much
will all the tiles weigh in kilograms?
9.
Molly needs two 2.7 m pieces and one 95 cm piece of wood trim for her doorway.
Wood trim costs $3.85 per meter plus 6.5% sales tax. How much will it cost Molly to
frame her doorway?
114
Bails – Math 094 Notes
Section 8.5 Metric-English Conversions and Temperature
Conversions between Systems
Length
Weight
Volume
2.54 cm = 1 in
1 m = 3.28 ft
1.61 km = 1 mi
28.3 g = 1 oz
2.2 lb = 1 kg
1.06 qt = 1 L
3.79 L = 1 gal
Make the following conversions by multiplying by appropriate conversion factors. If
necessary, round the answer to the nearest hundredth.
1.
35 gal = _____________ L
2.
2.75 kg = _____________ lb
3.
2.8 ft = _____________ cm
4.
600 m = _____________ yd
5.
15 qt = _____________ L
6.
12 kg = _____________ oz
7.
12.88 km =_____________ ft
8.
85 pt = _____________ L
115
Chapter 8 Measurement
9.
An economical dishwasher uses only 6.5 L of water. How many gallons does it use?
10.
The speed limit on the highway in Canada is 100 km per hour. What is the speed
limit in Canada in miles per hour?
11.
Phil went grocery shopping in Canada and the total bill was $154.25 in Canadian
dollars. The exchange rate is currently $1.00 US = $1.13 Canadian. How much was
the grocery bill in US dollars?
Temperature Formulas: Celsius C =
12.
116
22°C =_____________ °F
5
(F − 32)
9
or
13.
Fahrenheit F =
9
C + 32
5
122°F =_____________ °C
Bails – Math 094 Notes
Section 9.4 Rectangular Coordinate System
Paired Data – relationship between two numbers. For example: °C versus °F
This relationship is described as an Ordered Pair (x, y)
On the graph:
1.
Label the x-axis.
2.
Label the y-axis.
3.
Label the origin.
4.
The coordinate plane is divided into four
quadrants. Number the quadrants
The location of each point in a coordinate plane can be described by a pair of numbers called an
________________
________________, which is notated by (x, y). The first number, x,
represents the horizontal direction and the second number, y, represents the vertical direction.
5.
Label the points PQRS, where P is in
Quadrant I, Q is in Quadrant II, R is in
Quadrant III and S is in Quadrant IV.
6.
What are the coordinates of P?
7.
Which point has the coordinate ( −4,3 ) ?
8.
What are the coordinates of S?
9.
What are the coordinates of the point in
Quadrant III?
10.
Plot and identify the coordinates of one point that would lie between Quadrants I and II.
11.
Plot and identify the coordinates of one point that would lie between Quadrants I and IV.
117
Chapter 9 Rectangular Coordinate System
Plotting Points
Plot and label the points associated with each of the following ordered pairs.
12.
A (1,4 )
13.
B ( −2,3 )
14.
C ( −4, −1)
15.
D ( 3, −4 )
16.
E (1,0 )
17.
F ( 0, −3 )
18.
G ( −4,0 )
19.
H ( 0,5 )
Plot and label the points associated with each of the following ordered pairs.
20.
1⎞
⎛
J ⎜ 2, 3 ⎟
2
⎝
⎠
21.
K ( −3.2, − 4.7 )
22.
23.
24.
25.
⎛ 3 1⎞
L⎜ , ⎟
⎝4 2⎠
⎛ −9 ⎞
M ⎜ 2,
⎟
⎝ 2 ⎠
⎛ 2 ⎞
N ⎜ −3 , 0 ⎟
⎝ 3 ⎠
⎛ −17 ⎞
P ⎜ 0,
5 ⎟⎠
⎝
Complete each ordered pair with a number that will make the point fall in the specified
quadrant.
26.
Quadrant II
28.
Quadrant IV
118
( −2, _____ )
(12, _____ )
27.
Quadrant III
( _____, − 9)
29.
No Quadrant
( _____, 18 )
Bails – Math 094 Notes
Section 9.5 Introduction to Graphing
Plotting the points corresponding to the ordered pairs is called Graphing.
1.
Graph the points ( −1, 8 ) and ( 4, −2 ) and
draw a line through them
2.
Where does the graph cross the x-axis?
3.
Where does the graph cross the y-axis?
4.
What is one more point that would lie on
this line?
Steps to graphing a linear equation
1.
Find at least three ordered pairs that satisfy the equation.
2.
Plot the ordered pairs on a graph.
3.
Connect the points with a straight line.
4.
If the graph is not a straight line, re-check your work!
Graph the equations.
5.
x+y =3
x
y
( x, y )
−1
0
1
Two other solutions are
( _____, _____ ) and ( _____, _____ )
119
Chapter 9 Rectangular Coordinate System
Graph the equations.
6.
y = x −3
x
( x, y )
y
−1
0
1
Is the slope positive or negative?
7.
y = −2 x
x
( x, y )
y
−1
0
1
Is the slope positive or negative?
8.
y=
x
1
x
3
y
( x, y )
Is the slope positive or negative?
120
Bails – Math 094 Notes
Section 10.1 Product and Power Rule
Write each expression using exponents.
1.
3•3
2.
(−3) •(−3) •(−3)
3.
x•x•x•x•x•x
4.
⎛ x ⎞⎛ x ⎞⎛ x ⎞⎛ x ⎞
⎜ 2 ⎟⎜ 2 ⎟⎜ 2 ⎟⎜ 2 ⎟
⎝ ⎠⎝ ⎠⎝ ⎠⎝ ⎠
Write the following without exponents.
5.
32
6.
x3
7.
x2 • x3
8.
y2 • y
Identify the base and exponent for each expression.
9.
26
10.
( −3x )
8
The exponent is
The exponent is
The base is
The base is
PRODUCT RULE FOR EXPONENTS:
xm • xn = x m + n
When multiplying exponential expressions, keep the base and _____________ the exponents.
Use the product rule to simplify each expression.
11.
x 2 ix3
12.
73 i 7 4
13.
y2i yi y8
14.
( −5 ) ( −5 )
15.
( 2x )( 3 x )
16.
( −4 x )( 2x )
17.
( −5 x )( −6 x )
18.
( 3 x )( 4 x )( 2x )
3
6
10
4
2
6
8
6
121
Chapter 9 Rectangular Coordinate System
Expand the following then simplify.
19.
(x )
2
3
(x )
5
20.
POWER RULE FOR EXPONENTS: (x
2
m n
) = x m•n
m
EXPANDED POWER RULE FOR EXPONENTS:
(x
m
p n
y) =x
m•n
y
p•n
⎛x⎞
xm
and ⎜ ⎟ = m
y
⎝y⎠
When an exponential expression is raised to a power, keep the base and ___________________
all the exponents.
Use the power rule to simplify each expression.
21.
(x )
23.
( 3x )
25.
5 ( xy )
27.
⎛x⎞
⎜3⎟
⎝ ⎠
29.
( 2x y )
122
4
2
3
6
22.
(3 )
24.
( 5xy )
26.
⎛ 1⎞
⎜5⎟
⎝ ⎠
28.
( −3x y )
30.
( 6x y )
3
4
5
3
4
5
6
2
3
6
7
2
8
Bails – Math 094 Notes
Section 10.4 Adding and Subtracting Polynomials
A Monomial is a real number, a variable, or a product of real numbers and variables with whole
number exponents. Example: −5 , x, 5x , 7xy and 7x 2 .
A Polynomial is a monomial or a sum of monomials. Example: 3 x 3 + 2x 2 + 2x + 1
A Binomial is a polynomial with two terms. Example: 3 x − 4
A Trinomial is a polynomial with three terms. Example: 2x 2 + 5 x − 7
The Degree of a Term is the sum of the exponents on the variables.
The Degree of a Polynomial is the highest degree of any term in the polynomial.
Classify each polynomial according to the number of terms it contains (monomial,
binomial, or trinomial) then state the degree.
1.
3.
3x 2
2.
5x3 − 7x 2
Type:
Type:
Degree:
Degree:
−36
4.
6 x 4 − 8 x 2 + 10
Type:
Type:
Degree:
Degree:
Identify the coefficient of each monomial.
5.
−8x 2
6.
xy
Steps for Adding and Subtracting Polynomials
1.
Rewrite expression without parenthesis – distribute negative through parenthesis
2.
Identify like ______________
3.
Add/subtract the ____________________ (number in front of variable) & copy the variable
**Variables always stay the same in addition and subtraction
4.
Write answer in ________________________ order (highest power to lowest power)
Simplify the following and write the answer in descending order.
4y 3 − 5y 3
7.
8.
−15 x 2 + 3 x 2
123
Chapter 9 Rectangular Coordinate System
Simplify the following and write the answer in descending order.
9.
10.
x 2 – 3 x + 2x 2
4x 3 − x 2 – 5x 2 + 4x 3
Use the Distributive property on the following.
–1( 2x – 5 )
12.
11.
– –3 x 2 + 5 x + 10
Perform the indicated operations.
2x 2 − 3x + 4x 2 − 7 x
13.
14.
(7y
2
+ 8 y − 9 + ( −12y + 2 )
16.
(5x
3
+ 2x 2 + x − 3x 2 − x + 6x 3
Evaluate each polynomial when x = −2
17.
3 x 2 − 4 x − 10
18.
x3 − 5x 2 + 3x
If possible, simplify the following.
19.
7x + 7x
20.
7xi 7x
(
15.
( 4x
) (
4
) (
)
+ 2x 3 − 3 x 3 − 9 x 4
)
(
)
)
) (
21.
8x 2 + 9x
22.
8x 2 i 9x
23.
( 6x y )
24.
6x 2 + 5y 2 − 7x 2 + 3y 2
124
3
4
2
)