Unit 5: Equations and Inequalities

1
Unit 5 Equations and Inequalities

• Test Review

2

• What are like terms?
• Like terms are terms that have the same
  variables raised to the same power or exponent.
• Examples 2x and 5x, 2y2 and -6y2
• Simplify the following expression
• (2x - 6xy 3y) (-3x 4y 3xy)

-x
-9xy
7y
3

• 2. A rectangle has an area of 12x 18. What
  could the length and width of the rectangle be?
• Select all that apply.
• A. 2 and 6x 9
• B. 2x and 9
• C. 3 and 4x 6
• D. 6 and 9x
• E. 6 and 2x 3
• F. 4 and 3x 6

6x
9
A.
Correct
2
12x
18
2x
B.
9
18x
Incorrect
4

• 2. A rectangle has an area of 12x 18. What
  could the length and width of the rectangle be?
• Select all that apply.
• A. 2 and 6x 9
• B. 9 and 2x
• C. 3 and 4x 6
• D. 6 and 9x
• E. 6 and 2x 3
• F. 4 and 3x 6

4x
6
C.
Correct
3
12x
18
9x
D.
6
54x
Incorrect
5

• 2. A rectangle has an area of 12x 18. What
  could the length and width of the rectangle be?
• Select all that apply.
• A. 2 and 6x 9
• B. 9 and 2x
• C. 3 and 4x 6
• D. 6 and 9x
• E. 6 and 2x 3
• F. 4 and 3x 6

2x
3
E.
6
12x
18
Correct
3x
6
F.
Incorrect
4
12x
24
6

• 3. Which of the following expressions is/are
  equivalent to 12x 16 4?
• 4(3x 4 1)
• 4(3x 3)
• 12x 12
• 24x
• 28x 4
• 4(3x 4) 4

Start by simplifying the expression given 12x
16 4
12x 12
Then simplify each answer choice to see if it is
the same as the given expression.
7

• 3. Which of the following expressions is/are
  equivalent to 12x 16 4?
• 4(3x 4 1)
• 4(3x 3)
• 12x 12
• 24x
• 28x 4
• 4(3x 4) 4

12x 12

• 4(3x 4 1)
• 12x 16 4
• 12x 12

Correct
Correct
B. 4(3x 3) 12x 12
8

• 3. Which of the following expressions is/are
  equivalent to 12x 16 4?
• 4(3x 4 1)
• 4(3x 3)
• 12x 12
• 24x
• 28x 4
• 4(3x 4) 4

12x 12
C. 12x 12
Correct
Incorrect Cant combine unlike terms
D. 24 x
9

• 3. Which of the following expressions is/are
  equivalent to 12x 16 4?
• 4(3x 4 1)
• 4(3x 3)
• 12x 12
• 24x
• 28x 4
• 4(3x 4) 4

12x 12
Incorrect Cant combine unlike terms
E. 28x 4
Correct
F. 4(3x 4) 4 12x 16 4 12x 12
10

• 4. Megan, Heather, Brianne, and Rachel are
  sharing the cost of renting an apartment.
• Megan will pay 30 of the cost
• Heather will pay 25 of the cost
• Brianne will pay .20 of the cost
• Rachel will pay the remainder of the cost
• The cost of the apartment is 1250 per month.
  Fill in the table
• to show how much each person will pay each month.

Megan Find 30 of 1250
Megan
Heather
Brianne
Rachel
375
(.3 )(1250) 375
11

• 4. Megan, Heather, Brianne, and Rachel are
  sharing the cost of renting an apartment.
• Megan will pay 30 of the cost
• Heather will pay 25 of the cost
• Brianne will pay .20 of the cost
• Rachel will pay the remainder of the cost
• The cost of the apartment is 1250 per month.
  Fill in the table
• to show how much each person will pay each month.

Heather Find 25 of 1250
Megan
Heather
Brianne
Rachel
375
(.25 )(1250) 312.50
312.50
12

• 4. Megan, Heather, Brianne, and Rachel are
  sharing the cost of renting an apartment.
• Megan will pay 30 of the cost
• Heather will pay 25 of the cost
• Brianne will pay .20 of the cost
• Rachel will pay the remainder of the cost
• The cost of the apartment is 1250 per month.
  Fill in the table
• to show how much each person will pay each month.

Brianne Find .20 (or 20) of 1250
Megan
Heather
Brianne
Rachel
375
312.50
(.2)(1250) 250
250
13

• 4. Megan, Heather, Brianne, and Rachel are
  sharing the cost of renting an apartment.
• Megan will pay 30 of the cost
• Heather will pay 25 of the cost
• Brianne will pay .20 of the cost
• Rachel will pay the remainder of the cost
• The cost of the apartment is 1250 per month.
  Fill in the table
• to show how much each person will pay each month.

Rachel Set up an equation to find the amount
Rachel owes.
Megan
Heather
Brianne
Rachel
375
375 312.50 250 x 1250
312.50
937.50 x 1250 -937.50 -937.50
x 312.50
250
312.50
14

• 5. A shirt costs x dollars. A 7 sales tax must
  be added to the cost of the shirt. Kyle wants to
  multiply the cost of the shirt by 0.07 to find
  the tax and then add it to the cost of the shirt.
  Jeremiah thinks the cost of the shirt should be
  multiplied by 1.07. The expressions for the two
  methods are shown below.
• Kyle x 0.07x
• Jeremiah 1.07x
• Are the two expressions equivalent? Explain.
• What does this mean in the terms of the methods
• outlined by Kyle and Jeremiah?

Kyles method shows the shirt (x) added to the
tax amount (0.07x), which will give the total
price of the shirt with tax.
15

• 5. A shirt costs x dollars. A 7 sales tax must
  be added to the cost of the shirt. Kyle wants to
  multiply the cost of the shirt by 0.07 to find
  the tax and then add it to the cost of the shirt.
  Jeremiah thinks the cost of the shirt should be
  multiplied by 1.07. The expressions for the two
  methods are shown below.
• Kyle x 0.07x
• Jeremiah 1.07x
• Are the two expressions equivalent? Explain.
• What does this mean in the terms of the methods
• outlined by Kyle and Jeremiah?

Jeremiahs method shows that we are paying 107
of the price of the shirt, or 100 (cost of the
shirt) plus 7 (tax). This also gives us the
total price of the shirt including tax.
16

• 5. A shirt costs x dollars. A 7 sales tax must
  be added to the cost of the shirt. Kyle wants to
  multiply the cost of the shirt by 0.07 to find
  the tax and then add it to the cost of the shirt.
  Jeremiah thinks the cost of the shirt should be
  multiplied by 1.07. The expressions for the two
  methods are shown below.
• Kyle x 0.07x
• Jeremiah 1.07x
• Are the two expressions equivalent? Explain.
• What does this mean in the terms of the methods
• outlined by Kyle and Jeremiah?

Both methods are equivalent. You can see this by
combining like terms in Kyles expression x
.07x 1.07x This is Jeremiahs expression.
17

• 5. A shirt costs x dollars. A 7 sales tax must
  be added to the cost of the shirt. Kyle wants to
  multiply the cost of the shirt by 0.07 to find
  the tax and then add it to the cost of the shirt.
  Jeremiah thinks the cost of the shirt should be
  multiplied by 1.07. The expressions for the two
  methods are shown below.
• Kyle x 0.07x
• Jeremiah 1.07x
• Are the two expressions equivalent? Explain.
• What does this mean in the terms of the methods
• outlined by Kyle and Jeremiah?

This means that Kyle and Jeremiah can each use
their own method to solve the problem and they
will both reach the same answer.
18

• 6. For options A-E, choose all of the expressions
  that are equivalent to
• 2(4x 2).
• A. 2(2 4x)
• B. 8x 2
• C. 4x 2 4x 2
• D. 8x 4
• E. 6x 4

Simplify the given expression AND expressions in
each answer choice. Then compare. 2(4x 2) 8x
4

• 2(2 4x) 4 8x
• Equivalent (Commutative Property)

B. 8x 2 is NOT Equivalent to 8x 4

• 4x 2 4x 2 8x 4
• Equivalent

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• 6. For options A-E, choose all of the expressions
  that are equivalent to
• 2(4x 2).
• A. 2(2 4x)
• B. 8x 2
• C. 4x 2 4x 2
• D. 8x 4
• E. 6x 4

Simplify the given expression AND expressions in
each answer choice. Then compare. 2(4x 2) 8x
4
D. 8x 4 Equivalent
E. 6x 4 is NOT Equivalent to 8x 4
20

• 7. Sam is opening a ski rental business. He does
  not know how much equipment he needs to purchase,
  but he does know the cost of the equipment. This
  is shown in the table below.
• Which of the following expressions represent(s)
  Sams
• total cost to purchase his ski equipment?
• A. 60(x y) 30z
• B. 60 x 60 y 30 z
• C. 60(x y 30z)
• D. 60x 60y 30z
• E. 60(x y z 30)
• F. 60(x y z) 30z

First, write an expression matching the
situation 60x 60y 30z
Item Cost Amount to Purchase
Skis 60 x
Boots 60 y
Poles 30 z
Simplify each answer choice and compare to above
expression.

• 60(x y) 30z 60x 60y 30z
• Equivalent

B. 60 x 60 y 30 z 150 x y z
Not Equivalent
21

• 7. Sam is opening a ski rental business. He does
  not know how much equipment he needs to purchase,
  but he does know the cost of the equipment. This
  is shown in the table below.
• Which of the following expressions represent(s)
  Sams
• total cost to purchase his ski equipment?
• A. 60(x y) 30z
• B. 60 x 60 y 30 z
• C. 60(x y 30z)
• D. 60x 60y 30z
• E. 60(x y z 30)
• F. 60(x y z) 30z

First, write an expression matching the
situation 60x 60y 30z
Item Cost Amount to Purchase
Skis 60 x
Boots 60 y
Poles 30 z
Simplify each answer choice and compare to above
expression.
C. 60(x y 30z) 60x 60y 1800z Not
Equivalent
D. 60x 60y 30z Equivalent (same as above)
22

• 7. Sam is opening a ski rental business. He does
  not know how much equipment he needs to purchase,
  but he does know the cost of the equipment. This
  is shown in the table below.
• Which of the following expressions represent(s)
  Sams
• total cost to purchase his ski equipment?
• A. 60(x y) 30z
• B. 60 x 60 y 30 z
• C. 60(x y 30z)
• D. 60x 60y 30z
• E. 60(x y z 30)
• F. 60(x y z) 30z

First, write an expression matching the
situation 60x 60y 30z
Item Cost Amount to Purchase
Skis 60 x
Boots 60 y
Poles 30 z
Simplify each answer choice and compare to above
expression.
E. 60(x y z - 30) 60x 60y 60z -1800
Not Equivalent
F. 60(x y z) 30z 60x 60y 60z 30z or
60x 60y 30z Equivalent
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• 8. For options A-E, choose all of the expressions
  that are equivalent to
• 2(4x 2).
• A. 2(2 4x)
• B. 8x 2
• C. 4x 2 4x 2
• D. 8x 4
• E. 6x 4

Simplify the given expression AND expressions in
each answer choice. Then compare. 2(4x 2) 8x
4
D. 8x 4 Equivalent
E. 6x 4 is NOT Equivalent to 8x 4
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• 8. The scale below has 1-ounce weights and
  weights of x ounces.
• Write an equation that represents this situation
  
• Determine the value of x. Explain your reasoning.

3x 3 12
The value for x is 3.
First, you can take 3 of the 1 ounce weights off
of both sides and end up with 3x 9. Then, you
can divide and see that for each x, there will be
3 of the 1 ounce weights.
25

• 9. Four students are planning a trip. Each
  student needs at least 2000 for the trip. Each
  student writes an inequality that can be used to
  determine the number of weeks, x, that the
  student will need to save up the money needed for
  the trip. The inequalities are shown in the table
  below.

Which of the following students wrote an
inequality with a solution of 80? Select all
that apply.
Solve each inequality to find the matching
solution.
A. Kelly B. Anthony C. Sophia D. Troy
Kellys inequality DOES match 80.
-1200 -1200
10 10
26

• 9. Four students are planning a trip. Each
  student needs at least 2000 for the trip. Each
  student writes an inequality that can be used to
  determine the number of weeks, x, that the
  student will need to save up the money needed for
  the trip. The inequalities are shown in the table
  below.

Which of the following students wrote an
inequality with a solution of 80? Select all
that apply.
Solve each inequality to find the matching
solution.
A. Kelly B. Anthony C. Sophia D. Troy
Anthonys inequality DOES NOT match 80.
-200 -200
150 150
27

• 9. Four students are planning a trip. Each
  student needs at least 2000 for the trip. Each
  student writes an inequality that can be used to
  determine the number of weeks, x, that the
  student will need to save up the money needed for
  the trip. The inequalities are shown in the table
  below.

Which of the following students wrote an
inequality with a solution of 80? Select all
that apply.
Solve each inequality to find the matching
solution.
A. Kelly B. Anthony C. Sophia D. Troy
Sophias inequality DOES match 80.
-400 -400
20 20
28

• 9. Four students are planning a trip. Each
  student needs at least 2000 for the trip. Each
  student writes an inequality that can be used to
  determine the number of weeks, x, that the
  student will need to save up the money needed for
  the trip. The inequalities are shown in the table
  below.

Which of the following students wrote an
inequality with a solution of 80? Select all
that apply.
Solve each inequality to find the matching
solution.
A. Kelly B. Anthony C. Sophia D. Troy
Troys inequality DOES NOT match 80.
-1000 -1000
250 250
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• 10. Mrs. Rogers is saving up for a family
  vacation. She knows that she needs at least 2400
  and has already saved up 1200. She decides to
  set aside 60 every week. The inequality
  60w12002400 represents this inequality.
• Solve for w.
• Explain what this inequality means in terms of
  the family vacation.

-1200 -1200
60 60
The inequality shows that it will take at least
20 weeks to save enough money for the family
vacation.