Name:_________________________________________________ Period:____________
AAT
Algebra Flashback
Mr. DeGroh
Mrs. Grunloh
Mrs. Sokolowski
1
2
Expressions, Equations & Formulas
What is the order of operations? Hint: Think PEMDAS
Ex. 1: Simplify the expression.
384  3(7  2)   3
3
Ex. 3: Evaluate:
Ex. 2: Evaluate.
1
x2 y
x

5,
y


2,
z


if
4
xz
4
2
 2 y  1 when y  
3
3
Ex. 4: Formula for the Area of a trapezoid A 
1
h(b1  b2 ) where h represents the height, and b1 and b2
2
are the bases of the trapezoid. Find the area of the trapezoid with h = 10 in., b1 = 16 in., and b2 = 52 in.
3
Ex. 5: Simplify the expressions.
a. 4(6c  3d )  5(2c  4d )
b.
1
1
(10a  15b)  (8b  4a)
5
2
Ex. 6: Solve the equations.
a. 3(2a  3)  15  4(3a  6)
1
b. 5(c  8)  (2c  12)  23
4
c.
3
2
y  y5
4
3
Ex. 7: Simplify the polynomial expression.
a. (6x 2  7x  8)  (4x 2  9x  5)
b. (x 2  3x  4)  (4x 2  2x  9)
4
c.
4 2
x (6x 2  9x  12)
3
d. 5xy (2x  y )  6y 2 (x 2  6)
e. (3x  1)(2x 2  4x  5)
Ex. 8: Solve each equation or formula for the specified value.
a. 8r  5q  3 for q
b. S   rl   r 2 . Solve for l .
Ex. 9: Solve. Remember you need to get the absolute value ALONE first.
a. 3 2 x  3  5  4
b. 2 5 y  1  10
c. 2 3a  6  0
5
Expressions, Equations & Formulas Practice #1
1. Evaluate the expression.
x2  y( x  y)
if
x  8 and
Evaluate each expression if a 
3. ab 2  d
2. Evaluate.
y
3
,
4
8 xy  z 3
y2  5
3
2
b  8,
4.
c  2,
d  3,
ab
d 2
c
g 
if
x  5, y  2, z  1
1
. All answers will be nice numbers.
3
5. g 2 (b  dg )
6. The formula h  120t  16t 2 gives the height h in feet of an object t seconds after it is shot upward
from Earth’s surface. What will the height of the object be after 6 seconds?
Simplify:
7. 2(5 g  6h )  9(4h  2g )
8.
1
2
(6x  9y )  (10y  5x )
3
5
6
5 2
x (2x 2  4x  12)
2
9. (6x  11y 2  6)  (4  7y 2  x )
10.
11. 3a (ab 2  3a )  5a 2 (b 2  2b )
12. (7x 2  4x  3)(5x  2)
9
.
5
b 3c  4a
Evaluate each expression if a  3, b  5, c 
13.
b 2  2c
14.
15.
4  2b  10 c
Solve the equations.
16. 5( 2x  4)  3(4 x  5)  97
17.
2
3
(6c  18)  (8c  32)  18
3
4
18.
4
2
y 5  y 5
9
3
7
19. z  13  21
1
3
20. 5 w  9  20
Solve for the indicated variable.
21. Solve E  mc 2 , for m.
22. Solve
x y
 3 , for x.
z
23. The equation for the volume of a right circular cylinder is V   r 2h . The volume can be found if you
know the height and radius of the cylinder. Solve the volume equation for r.
24. Mrs. Grunloh and Mrs. Sokolowski decide to go play a game of golf. Mrs. Sokolowski needed to rent
clubs for $6. The total bill for two rounds of golf and one club rental was $76. How much is one round
of golf? Write an equation to represent the scenario and solve it.
8
Expressions, Equations & Formulas Practice #2
2
33 - ( 8 ÷ 2 + 42 )  - 25

1. 
16
1
2
2.
A carpenter charges $10 per
square foot to lay a floor. If a
square-shaped hallway is 6
feet along one side, and the
customer has a coupon for
$25 off the total, then how
much will the floor cost?
Evaluate each expression for the r  1, n  3, t  12, & w  
4. nr 2  wt
5. 7 n  2t 
2w
r
3.
A caterer charges a setup fee
of $50, plus $20 per person.
How much will the caterer
charge if 35 people attend
the party, and the customer
has a coupon for $100 off the
total?
1
2
6.
9r 2  (n 2  1)t
 w t  (t  r ) 
Simplify each expression
7. 2a3 (10m  7a)  3m(a3  2m)
2
8. 17 x y  14 
1
xy (2 x  50)
2
9. (2x  y )(x 2  6xy  3y 2 )
9
Solve equations using the properties of equality
10.
3
x  5 x  2  14
5
11. 10(1 2y)  5(2y 1)
12.
1
2
(8  12 x)  15  5 x   18
4
5

Solve each formula equation for the indicated variable
13. Solve for r : I  prt
14. Solve for h : a  gh  qw
15. Solve for d : r 
d
t
Solve equations involving absolute values
16. 4 3y  2  8  0
17. 2 x  2  4
10
Inequalities
Key Idea:_______________________________________________________________________
______________________________________________________________________________
Symbols:
Ex.1: Graph the inequality on a number line. Express your answer in interval notation.
a. x  3
b. x  2
Interval Notation:_________________
Interval Notation:_________________
Solve and express your answer in interval notation.
Ex. 2:

5
 x  12   21
6
Interval Notation:_________________
Ex. 4: Solve  m 
m4
9
Interval Notation:_________________
Ex. 3: Solve
7
5
1
x   6x 
2
3
6
Interval Notation:_________________
Ex. 5: Solve 3(a  4)  2(3a  4)  4a  1
Interval Notation:_________________
11
Ex. 6: Write an inequality for the situation & solve.
At Game-O-Rama video game rental company you can rent games for $3 each. There is a one-time
$4.50 sign-up fee for the company. You receive a $50 gift card for this company, at most how
many games can you rent?
a. Write an inequality to represent the situation & solve
b. At most how many games can you rent?
Inequality Practice: Solve each inequality, graph your solution, and write your final answer in interval
notation.
1. 8x 
3
 10
2
2. 
3. 9x  11  6x  9
16 8
 r 0
3 5
4. 9(2r  5)  3  7r  4
1
2
5. 36  2(w  77)   (2w  52)
6. The Grunloh Hotel cost $90 per night plus $12
per day for parking. You can also except to pay $30
in tips during your stay. Write and solve an
inequality to find out how many nights you can stay
at the Grunloh Hotel without exceeding $600.
12
Compound Inequalities
“And” Compound inequalities
x  1 and x  2
Another way to write x  1 and x  2 is:
Interval Notation:
“Or” Compound Inequalities
x  1 or x  4
Interval Notation:
Ex. 1: Graph each solution set on the number line. Then write your solution in interval notation.
a.
x  2 or x  1
b.
x  5 and x  2
Interval Notation:
Interval Notation:
c.
d.
x  4 and x  1
Interval Notation:
e.
1  x  3
Interval Notation:
x  0 or x  2
Interval Notation:
13
Ex. 2: Solve the following “AND” compound inequalities. Graph & state in interval notation.
a. 3  x  1  2
b. Twice a number increased by 7 is
between 13 and 17.
Interval Notation:
Interval Notation:
Ex. 3: Solve the following “OR” compound inequalities. Graph & state in interval notation.
a. 3x  7  2 or 3x  7  2
b. Two times a number decreased by 3 is
greater than 15 or three minus the product of 7
and a number is less than 17.
Interval Notation:
Interval Notation:
Ex. 4: Solve the AND or OR compound inequalities. Graph & state interval notation.
a.
15 5
 x  0 and 5x  6  14
2 3
Interval Notation:
b.
7
1
4 4
(5x  2)  or 6x    5x
8
8
3 7
Interval Notation:
14
Compound Inequality Practice: Solve each inequality, graph your solution, and write your final answer in
interval notation.
“And” means____________________________
1. 10  3x  2  14
3.
1
20
7
x

2
3
3
5. 3y  2  7 or
and
2
x 5 3
5
2y  1  9
7. 5(w  4)  5 and
1
(w  4)  12
2
9. Six times a number minus 2 is between 22 and
82.
“Or” means____________________________
2. 18  4x  10  50
4. 5x  2  13 and
4x  1  19
6. 4d  1  9 or
1
1
d  2
4
3
8. 3(6  y )  6 or
6y  8
10. Three times a number increased by 20 is
greater than or equal to 32 or negative eight times
that same number increased by one is less than 17.
15
Absolute Value Inequalities
If a  b then b  a  b
“Less thand”
If a  b then a  b
or
a  b
“Greator”
Ex 1: Solve the absolute value inequality. Graph your solution. State answer in interval notation.
a. 2x  2  6
b. 10  2x  2
Interval Notation:
Interval Notation:
c. 2
5
b  4  12
3
Interval Notation:
d.
x
5
 5   10
2
2
Interval Notation:
16
Absolute Value Inequality Practice
1. Solve each absolute value inequality.
If a  b then b  a  b
2. Graph your solution. Do NOT skip this step!
If a  b then a  b or
3. Write your final answer in interval notation.
1. 4k  1  27
3.
x
 5  2  10
2
5. The depth of an aquarium tank for dolphins is
represented by d  50  5 . What is the range
the depth can be?
2.
“Less thand”
a  b
“Greator”
5
w  2  28
3
4. 4
1
2
p   8
3
3
6. Find the error: Sabrina and Isaac are solving
3x  7  2 .
Sabrina:
3x  7  2
3x  7  2 or 3x  7  2
3x  5 or 3x  9
x 
5
or x  3
3
Isaac:
3x  7  2
2  3x  7  2
9  3x  5
3  x 
5
3
17
Quiz Review on Expressions, Equations, Formulas and Inequalities
SHOW ALL WORK.
Use the order of operations and the properties of real numbers to evaluate and simplify expressions.
1
8 3  (10  14) 2
1. Simplify: 2 5  13 
9
2
2. Evaluate: r  5r  7 for r  3


w3  1
3. Evaluate:
for w  3
w6
4. Evaluate: ab 
2a
1
when a  12 & b 
4
b
5. You and 4 friends went out to dinner. The group ordered 2 appetizers for $6.99 each, 5 sodas for
$2.19 each, 3 pizzas for $12.50 each. The tax came to $4.53. Find out how much each of you owe
for the bill?
6. Simplify:
13 2
1
x  19  x(1  3 x)
2
2
7. Simplify (3x  y )(2x 2  4xy  7y )
Solve each equation.
8.
3( w  5)  2( w 
17
)
4
9.
5(a  4)  13  3(a  7)
10.
1
1
(10 x  15)  4(2 x  5) 
5
2
18
Solve each formula equation for the indicated variable
11.
2
Solve for h : V  r h
12. Solve for y: 5 y  6 x  7
 
Solve each inequality and graph the solutions. Then write the solutions in interval notation.
13.
3
2 y  1  4( y  )
2
I. N. _________________
14. 7 y  35  2 y
I. N. _________________
Solve each compound inequality and graph the solutions. Then write the solutions in interval notation.
15.
3  2x  3  5

16.
2x 
1
 5 or 3x 10  x
5

I.N._________________
I.N._________________
19
Solve each equation and check for extraneous solutions.
17.
2
x 9 1
3
18.
1
4 5  x  1  15
2
19.
2 x  9  8

Solve each inequality and graph the solutions. Then write the solutions in interval notation.
20.
5
2 x  20
2
21.
5 3x  7  4  29

I.N._________________
I.N._________________
20
Multiple Choice:
________22. Solve & write solution in interval notation.
2x  3  7
A. (,5]
B. [5,5]
C. (, 2] [5, )
D. [2,5]
________23. Select the algebraic expression that represents the verbal expression.
Twice the sum of a number and 8..
A.
B.
C.
D.
2n + 8
n + 16
2(n + 8)
2n + 28
_________24. Solve 4(2 x  9)  3x  4
A. -32
32
B. 
5
40
C.
3
D. 8
________25. Evaluate  2a  b if a  4 & b  3
A.
B.
C.
D.
-5
5
-8
8
________26. Solve. Write answer in interval notation.
8 x  4  12
A. (, 1)
B. (, 2)
C. (1, )
D. (2, )
21
Lines
Function Notation:________________________________________________________________
Ex. 1: Find the values of the following given f (x )  x 3  3 and g (x )  0.3x 2  3x  2.7
a. f(2)
b. f(-3)
c. g(4) + 9
d. 5g(-2)
e. f(1) + g(2)
f. 7g(3) - 10
Slope Formulas:
Ex 2: Find the slope of the line that passes through each pair of points.
a. (-1, 4) & (3, -8)
b. (5, 3), (-4, 3)
c. (-6, 4) & (-6, 2)
Ex 3: Find the slope of each line shown below.
a.
b.
22
Ex. 4: In 2008 a Nintendo DS lite cost $130. In 2010 it now costs $100. Find the average rate of
change.
Ex. 5: In May, Glen sent 658 text messages. Later that year in September, he sent 874 text messages.
Find the average rate of change.
Ex. 6: The “slope” of a road can be defined as the “rise” divided by the “run”. Use the information below
to find the slope of the two roads. Then use the information to determine which road is steeper.
a. In road A, for every 50 feet horizontally the road rises a height of 30 feet. Determine
the slope.
b. In road B, for every 50 feet horizontally the road rises a height of 10 ft. Determine the slope.
c. Which road is steeper?
Different Forms of Linear Equations:
1._________________________________ 2.___________________________________
Ex. 7: Write an equation of the line show in the graph in slope-intercept form.
a.
b.
______________________________
______________________________
23
c.
d.
______________________________
______________________________
Ex. 8: Using the given information, graph the line.
a. 3x  8y  40
b.
2
1
x  y 1
3
4
Ex. 9: The Geek Squad charges a flat fee of $50 to come and assess your computer problem plus a $25
per hour fee to fix the problem.
a. Write a linear equation in slope intercept form to represent this situation for total cost (y) in
terms of the number of hours (x) needed to fix the computer.
b. Use the equation to calculate the how many hours it took to fix your computer if the cost
$225.
24
Ex. 10: The median price of an existing home in the United States was $139,000 in 2000. The median
price had risen to $191,300 by 2004. Find the average annual rate of change in median home price from
2000 to 2004.
Ex. 11: The SheiKra roller coaster at Busch Gardens in Tampa, Florida, features a 138-foot vertical
drop.
a. What is the slope of the coaster track at this part of the ride? Explain.
b. Write a linear equation in slope-intercept form to represent this situation.
Ex. 12: The graph shows the annual coal exports from U.S. mines in millions of short tons.
a. What was the rate of change in coal exports between 2001 and 2002?
b. How does the rate of change in coal exports from 2003 to 2004 compare to that of 2001 to
2002?
c. Explain the meaning of the part of the graph with a slope of zero.
25
Parallel vs. Perpendicular Line:
1. If the ___________________ of lines are _____________________, then the lines are
_______________________________.
2. If the ___________of lines are ___________________ & __________________, then the lines
are ___________________________________.
Ex 13: Write an equation in slope-intercept form that satisfies each set of conditions.
a. passes through (4, 1) & parallel to
y  3x  5
b. passes through (-5, 7) & perpendicular to
y  5 x  3
c. passes through (4, -10) & parallel to a line with a slope of m 
7
.
8
5
d. passes through (-9, -3) & perpendicular to y   x  8
3
26
e. parallel to x-axis passing through (4, -3)
f. parallel to the y-axis passing through the point (-2, 3)
g. perpendicular to the line y = -3 passing through the point (5, -8)
h. perpendicular to 5x  2y  8 passing thru (-3, 5)
27
Lines Practice
Directions: Given
1. f (10)
f (x )  3x  5 &
g ( x )  x 2  4x
2. g (2)
3. f (1)  g (5)
Find the slope of the line that passes through each pair of points.
5. (-2, 11), (5, 6)
6. (8, 2), (8, -100)
7.
4. 3f (4)  5
8.
Using the given information, graph the line.
9. 4x 
2
y  16
3
10.
3
x 3
5
11. In 2006 the cost of a desktop computer was $860. In 2010 a desktop computer costs $376. Find
the average rate of change.
28
12. After 6 days you made 11 tweets on Twitter and after 9 days you made a total of 25 tweets, find
your average rate of change between days 6 and 9.
13. The surface of Grand Lake is at an elevation of 648 feet. During the current drought, the water level
is dropping at a rate of 3 inches per day. If this trend continues, write an equation that gives the
elevation in feet of the surface of Grand Lake after x days.
14. Wade’s grandmother gave him $100 for his birthday. Wade wants to save his money to buy a new
MP3 player that costs $275. Each month, he adds $25 to his MP3 savings. Write an equation in slopeintercept form that represents the number of months that it will take Wade to save $275.
.
15. Find the equation of the line that passes through (2, -4) and is perpendicular to the line y  2x  5 .
16. Find the equation of the line parallel to x  3y  7 through (6, 5).
17. Write the equation of the line that is perpendicular to the x-axis and passes through (-5, -6).
29
Quiz Review on Lines
Find the values of the following functions.
f ( x)  2 x2  x
1. f (3)
g ( x)  3x  1
1
2. g  
2
3. h(4)
h( x)  ( x  1)2
4. 3 f (2)  2 g (2)  1
Find the slope from the give information.
5. (-3, 5) & (2, -10)
6.
7. (5, -2) & (5, 0)
8. Emma earned $126 in 3 weeks of babysitting. After babysitting for 8 weeks she has earned $336.
Find the average rate of change.
9. In the year 2007, GNHS had 346 in the graduating class. In 2010 the graduating class was 544.
What is the average rate of change.
Write the equation of the line in slope-intercept form from the given information.
10.
11.
30
Graphing equations of lines from given information.
12.
y4
1
 x  5
5
13. y  2
14.
1
3 1
y   x
4
4 2
15. Henry started a new job in which he is paid $9.50 an hour. Write and solve an equation to determine
Henry’s gross salary for a 40-hour work week.
16. The engine of a chainsaw requires a mixture of engine oil and gasoline. According to the directions,
oil and gasoline should be mixed as shown in the graph below.
a. Write an equation to help you determine the ratio of oil to
gasoline for any amount.
b. If 12 fluid ounces of oil are used, how much gasoline needs
to added?
31
17. Write an equation of the line parallel to y 
1
x  5 that passes through the point (-4, 1).
2
18. Write an equation of the line perpendicular to y 
1
x  5 passing through (-10, 5).
2
19. Write an equation of the line perpendicular to the y-axis passing through (-2,-3).
20. Write the equation of the line through (1, -7) that is parallel to a line passing through
the points (3, 4) and (-2, 9).
32
Systems
Systems Pre-Assessment:
Solve these systems by substitution.
1.
y  x 1
2.
2x  3y  12
4x  3y  6
x  3y  3
Solve the system by elimination
3.
2x  y  17
4.
3x  y  8
2x  3y  9
x y 2
Solve the system by graphing
5.
y  x  4
6.
y  x 2
Answer Bank:
(5, -7)
(3, -4)
(3, -2)
y  2x  2
x  3y  15
(3, 1)
(3, 2)
(3, 1)
33
Steps to Solving Application Problems:
1. Assign variables.
2. Set up 2 equations.
3. Solve using elimination or substitution.
4. Make sure to answer final question.
1) The sum of two numbers is 12. The difference of the same two numbers is -4. Find the numbers.
2) Twice a number minus the second number is -1. Twice the second number added to three times the
first number is 9. Find the numbers.
3) Last year the volleyball team paid $5 per pair of socks and $17 per pair of shorts on a total purchase
of $315. This year they spent $342 to buy the same number of pairs of socks and shorts because the
socks now cost $6 and shorts cost $18. Write a system of equations & solve for the number of socks &
shorts they bought.
4) The admission fee at a small fair is $1.50 for children and $4.00 for adults. On a certain day,
2200 people enter the fair and $5050 is collected. How many children and how many adults attended?
34
5) During the 1996-1997 National Basketball Association season, the Boston Celtics played 82
games. They lost 52 more games than they won. What was their win-loss record that year?
6) Josh and Langston found that the width of their basketball court was 44 feet less than the
length. If the perimeter was 288 feet, what were the length and the width of their court?
7) Julie wanted to frame several family photos, including some of her recent wedding. She went to a
discount store and purchased two 11 x 14 frames and three 8 x 10 frames costing $22 (before
taxes.) Later she returned to the store and purchased one 11 x 14 frame and two 8 x 10 frames costing
$13 (before taxes.) How much did Julie pay for each of the different sized frames?
8) Two small pitchers and one large pitcher can hold 8 cups of water. One large pitcher minus one small
pitcher constitutes 2 cups of water. How many cups of water can each pitcher hold?
9) A test has twenty questions worth 100 points. The test consists of True/False questions worth 3
points each and multiple choice questions worth 11 points each. How many multiple choice questions are
on the test?
35
System of Inequalities:
Solution:_______________________________________________________________________
Example 1:
1

 y  x 1
2

 y  x  3
Example 2:
 x  y  5

y  4
Example 4:
 x  y  2

 x  y  6
a. Is (5, 4) a solution?________________
b. Is (-1, 7) a solution?________________
Example 3:
x  3

 x  2
a. Is (1,-3) a solution?________________
b. Is (-3, -2) a solution?________________
36
Example 5:
x  0

 y  1
 y  x  4

Example 6: Can you write a system of
inequalities for the graph shown below?
Example 7: The sum of two numbers is less than 7. Each of the numbers must be greater than zero. The
first number is less than 5 and the second number is greater than 2. Find the feasible region that
represents the different possibilities for the numbers.
Example 8: The sum of Steve’s age and Jenny’s age is less than 30. Steve’s age is greater than Jenny’s
age. Jenny is older than 5 years old. Graph the feasible region for their ages.
37
Systems Equation and Inequalities Practice:
Solve the system using any method.
1.
x  2  3y
4x  12y  8
2.
4.
3x  4 y  12
2 x  3 y  9
5.
2x  2y  4
x  2y  0
y  4x  2
y  2x  4
3.
5x  2 y  1
y  1  3x
6.
3 x  6 y  12
2x  y  5
7. Twice a number minus a second number is -1. Three times the first number added to twice the second
number is 9. Find the two numbers.
8. Drew bought three bags of chips and two bottles of soda and paid $9.15. His friend, Charles bought
two bags of chips and four bottles of soda for $8.74. What is the price of a bottle of soda? What is the
price of a bag of chips?
9. The campus bookstore received two shipments from Apple Computer over the last month. The first
contained six iPods and eight MacBooks, and cost to the bookstore was $6840. The second shipment
was three iPods and five MacBooks, at a cost of $4,170. What is the cost for one iPod and what is the
cost for one MacBook?
38
10.
y  x2
y2
1
x 3
11.
4
 x  4 y  12
y
a. Is (0, 0) a solution?________________
b. Is (0, -2) a solution?________________
12.
y  3x
5 x  3 y  0
13.
y3
x4
a. Is (5, 4) a solution?________________
b. Is (-3, -5) a solution?________________
y5
14. x  3
15. Write a system of inequalities for the graph.
y  x4
39
Factoring
First:
3
How many terms
does the polynomial
have?
4
2
2
40
Type of Factoring
Practice Problem #1
Practice Problem #2
4x y  4x
10a 3b 2  15a 2b  5ab 3
4ab  8b  a  2
5c  10c 2  2d  4cd
x 2  2x  15
2x 2  10x  12
2x 2  5x  3
3x 2  2x  5
4x 2  49
2x 3  50x
3
GCF
Grouping
Trinomial
Trinomial with leading
coefficient greater than 1
Difference of Two
Squares
41
Factoring Practice
Factor completely. If not factorable, write Not Factorable. Show all work necessary! Hint: Always look for GCF first!
1.
4 x 2  16 x
2.
x4 y3  x3 y 2
1._____________________
2._____________________
3.
2 x  5 xy  7 y
4.
x 3  4 x 2  4 x  16
3._____________________
4._____________________
5.
3x 3  12 x 2  x  4
6.
2x3  x 2  2x  1
5._____________________
6._____________________
7.
x 2  6x  5
8.
x 2  4 x  32
7._____________________
8._____________________
9.
4 x 2  20 x  25
10.
x2  6 x  9
9._____________________
10.____________________
11.
2 x 2  3x  9
12.
4 x 2  15 x  4
11.____________________
12.____________________
13.
x 2  25
14.
2 x 2  32
13.____________________
14.____________________
15.
x2  9
16.
Sarah is building a frame for her
artwork. If her canvas has an area of
2 x2  7 x  6 ,
write expressions that represent
the length and width of the
picture.
15.____________________
16.____________________
42
Monomials
Monomial – Is an expression that is a number, a variable, or the product of a number and one or
more variables. Monomials cannot have denominators, negative exponents, or a variable under a
radical.
Monomials:
Not Monomials:
Product of Powers: For any real number a and integers m and n,

a ma n  a m  n .
Which means, when multiplying with same base, add the exponents.
2
9
2 9
 411
Ex. 4  4  4
2b 3  6b 5  2  6b 35  12b 8
Simplify:
a. y 7  y 3  y 2
b. t 9  2t 8
c. (2a 3b )(5ab 4 )
d. (3x 2y 4 )(4x 3y 2 )
Zero Property: For any real number raised to the zero power,
2x y 
Ex. x 0  1
Simplify:
3
a. 3x y
5
0
0
a 0  1.
3x 4 y 0  3x 4
b.
 5x 4 y 7

4
 3z



0
Quotient of Powers: For any real number a  0 , and integers m and n,

am
 a m n .
n
a
Which means, when dividing with same base, subtract exponents
3
5
53
 1  531  52
5
5
Ex.
Simplify:
x7
 x 7 3  x 4
3
x
a.
3m 4
12m
43
Negative Exponents: For any real number a  0 and any number n,
a n 
1
a
and
n
1
a
n
 an .
Which means, when there is a number to a negative exponent, make it positive by flipping into a fraction

1 1

23 8
23 
Ex.
1
b
8
Simplify:
a.
 b8
p3
p8
b.
Suppose a and b are real numbers
Properties of Power:
Power of a Power
(a m )n  a mn
Power of Product
(ab )m  a mb m
Power of a Quotient
a 
 
b 
n
(a 2 )3  a 6
Ex. (xy )2  x 2y 2
3
a 
a3
Ex.    3
b
b 
b0
n
b 
bn
   n,
a
a 
and m and n are integers.
Ex.
n
a 
an
   n,
b
b 
a 5b 4
a 2b 3
a  0,b  0
x 
Ex.  
y 
3
3
y 
y3
   3
x
x 
Simplify:
5
a. (b )
b. (3c d )
2 4
2
5 3
 2a 
c.  2 
 b 
Apply the concepts:
1. 3a  2a
2
 2a 5 
5. 

 5a 
5
8a 5
2.
4a 2
2
3 2
6. ab c
3.
x 
d.  
3
 3a 2 
8a 4 b 0
7.
12ab3
4
3a 
4 0
3
4.
25a
 4 x2 y5 
8. 
3 
 2 xy 
2
44
Monomial Practice
Simplify. Assume that no variable equals 0.
1. 2y 7  y 3  3y 2
4.
12m 8 y 6
5.
9my 4
 9s 0r 3t 2 
7. 
2 

 18rt

2.
3
8.
 2b
c3
2

3
3.
9xz 3

x  y
2
y 3 5xy 8
3
4
e 5f 4
2
 5de
 2x 3 y 2
6.  3 5
 3x y
27 x 3 yz 0
3x
 4d
2

9.
f 1
3

2



20(m 2v )(v 2 )3
5m 3v 4
45
46
Quiz Review for Systems, Factoring and Monomials
Solve the following systems of equations. Use ANY method
y  x  3

3x  y  1
2.
2x  2y  32
4. 
x  y  16
5.
1.
2x  3y  9

4x  2y  22
1

y  x  1
3. 
3
 y  4x  1

x  y  4

x  y  9.5
x  y  4
6. Graph the system of inequalities 
2y  x  6
7. Katie and Danny have at least 3 pets between the two of them. Katie has more pets than
Danny. Danny has less than 5 pets and Katie has less than 7 pets. Find the feasible region for their
number of pets.
47
8. Write a system of inequalities for the graph.
Translate & Solve the Systems
9. The sum of twice a number and three times a second number is 12. Four times the first number minus
five times the second number is 13. Find the numbers.
10. G&S Clothing store sells all tops for the same price and all pants for the same price. Jen purchases
three tops and five pants for $42. Beth goes in and purchases one top and two pairs of pants for $16.
How much does G&S charge for a top and for a pair of pants?
11. For an upcoming concert, one lawn seat and one general admission seat costs $38. You and four
friends go to this concert and buy two lawn seats and three general admission seats for $98. How must
does a general admission ticket cost?
12. You go into Cand-o-lious and buy 3 pounds of chocolate covered peanuts and 2 pounds of gummy bears
for $9.25. Your favorite math teacher goes into the candy store and buys 8 pounds of gummy bears and
one pound of chocolate covered peanuts for $23.25. How much is it for one pound of gummy bear? How
much is it for one pound of chocolate covered peanuts?
48
Factor the following:
13. x 2  3x  2
14. 4x 2  25
15. 12x 2y  6xy  18xy 3
16. 4x 2  4x  1
17. 2x 2  x  3
18. 3x 3  3x 2  60x
19. x 3  5x 2  2x  10
20. a 2  3ay  2ay 2  6y 3
Simplify the Monomials:
21. (5x 3y 4z 2 )3
9xz 4
24.
27x 3 (2xy )0
22.
25.
7 x 6 y 4
23. (5a 7b 2 )(4a 2b )
21x y
2
2x
3
y 2

6x 0 y 8
3
 4x 5 y 3 z
26. 
2 0 3
 16x y z
2



49
Algebra Flashback Test Review:
Part 1: Expressions, Equations, Formulas & Inequalities.
Simplify.
1.
(8)1/3  2(5  3) 2
81  (2)
2
2. 3a  4b  | c | if a  2, b  6, c  3
4. Simplify: 3x 2 y  2 xy  4 xy (12  3x)
3. 2m 
3
1
1
n m n
4
2
8
5. (3x  y )(2x 2  4x  7y )
Solve.
6. 3( a  4)  6  2( a  3)
7.
1
3
3
( x  7)   2 x  5 
3
4
5
8. Solve for a: A 
1
h( a  b)
2
Solve each inequality and graph the solution. Then write the solution in interval notation.
9.
y 5
1
y 3
4
I.N._________________________
10.
1
 (2 x  7)  5
3
I.N._________________________
50
Solve each compound inequality and graph the solution. Then write the solution in interval notation.
11.
3  5x  2  7
I.N._________________________
12.
2
x3 7
5
or
3
(6 x  4)  4 x  2
2
I.N._________________________
Solve.
13. 2 3x  5  18
14. 2 3x 
2
 16  12
3
Solve each inequality and graph the solutions. Then write the solutions in interval notation.
15. 2 x  5  8
I.N.__________________________
16. 3 3x  4  12  6
I.N._________________________
51
Part 2: Lines
Given:
f (x )  3 x 
3
4
&
17. f(2)=
g  x   5x  x2
1
2g    3 
2
19.
18. g(-3) + 4 =
20. Determine the slope of the line given the points (5,-2) & (-9,10).
21. The cost of a flat screen TV in 2012 costs around $1925. In 2008 a similar flat screen may have
cost you $3000. What is the rate of change in price?
22. Write an equation of a line that passes
1
through (1, 5) and has a slope of  .
3
24. Write the equation of the line:
23. Write the equation of the line with a
slope of 0 going through (-2,4).
25. Graph the line: 3x + 4y  8
26. Graph the line: m=0, (2,-4)
52
27. “DJ Entertainment” charges a set-up fee of $400 and $172 an hour for DJ-ing the dance.
a. Write an equation that gives the total cost, y, in terms of the number of hours, x.
b. If the total number of hours spent was 7, how much was charged?
c. If your bill was $1131, how many hours did you hire DJ Entertainment for?
28. Write an equation of a line that is parallel to the graph of 2x + 3y = 5 and passes through
the point (-12, 1).
29. Write the equation of the line perpendicular to y  3x  2 going through (2, -7).

30. Write an equation of a line is parallel to the x-axis and passes through the origin.
53
Part 3: Systems, Factoring & Monomials
31.
 y  3x  2

3x  y  8
Solution:_________________
32.
1

y   x 3
33. 
2
 y  1
 x  2 y  10

3x  4 y  5
Solution:_________________
Solution:_________________
Graph the system of inequalities.
34.
1

y  x  2
3

 y  2 x  2
35.
2

y  x 3
3

 x  3
36. The difference between two numbers is twelve. The sum of seven times the first number and three
times the second number is negative eleven. Find the two numbers that will make these statements true.
37. Three hundred and thirty people came to the play The Curious George. Student tickets cost $5.50
and adult tickets cost $8 and the ticket booth made $2190. How many of each ticket was sold?
54
38. Write the system of inequalities for the graph shown below.
Factor the following.
39. 4ab 2  12a 2b 3  8ab 3
40. 12x 2  27
41. x 2  5x  6
42. 6xy  8x  21y  28
43. 4 x 2  3x  1
44. 4x 2  20x  25
Simplify the monomials.
45.
2a b c  4a c 
3
4
5
0
46.
27 x 3 yz 2
3x
1
y4

3
 7 g 2h
47. 
 14 g 5h 3





2
55