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					Lesson 8-1 Solving Equations with Variables on
           Each Side
Lesson 8-2 Solving Equations with Grouping
           Symbols
Lesson 8-3 Inequalities
Lesson 8-4 Solving Inequalities by Adding or
           Subtracting
Lesson 8-5 Solving Inequalities by Multiplying or
           Dividing
Lesson 8-6 Solving Multi-Step Inequalities
Five-Minute Check (over Chapter 7)
Main Idea
Targeted TEKS
Example 1: Equations with Variables on Each Side
Example 2: Equations with Variables on Each Side
Example 3: Real-World Example
• Solve equations with variables on each side.
8.2 The student selects and uses appropriate
operations to solve problems and justify solutions.
(A) Select appropriate operations to solve
problems involving rational numbers and justify the
selections. (B) Use appropriate operations to solve
problems involving rational numbers in problem
situations. Also addresses TEKS 8.5(A).
                   Equations with Variables on Each Side

    Solve 5x + 12 = 2x. Check your solution.

          5x + 12 = 2x              Write the equation.
    5x – 5x + 12 = 2x – 5x          Subtract 5x from each side.
                 12 = –3x           Simplify.
                 –4 = x             Mentally divide each side by
                                    –3.


Subtract 5x from       Subtract 5x from
the left side of the   the right side of
equation to isolate    the equation to
the variable.          keep it balanced
            Equations with Variables on Each Side

To check your solution, replace x with –4 in the original
equation.
Check      5x + 12 = 2x         Write the equation.
                   ?
        5(–4) + 12 = 2(–4)      Replace x with –4.
                   ?
         –20 + 12 = –8          Simplify.

               –8 = –8         This statement is true.

Answer: The solution is –4.
                                            BrainPOP:
                                        Two-Step Equations
Solve 7x = 5x + 6. Check your solution.

A. –3


B.


C. 3
                                                 A. A
                                                 B. 0% B
D. 21                                  0%   0%             0%
                                                 C. C

                                   A




                                            B




                                                   C




                                                           D
                                                 D. D
           Equations with Variables on Each Side

A. Solve 7x + 3 = 2x + 23.

    7x + 3 = 2x + 23         Write the equation.
7x – 2x + 3 = 2x – 2x + 23   Subtract 2x from each
                             side.
    5x + 3 = 23              Simplify.
 5x + 3 – 3 = 23 – 3         Subtract 3 from each
                             side.
        5x = 20              Simplify.
         x=4                 Check your solution.
Answer: The solution is 4.
            Equations with Variables on Each Side

B. Solve 1.7 + a = 3.8a – 5.3.
    1.7 + a = 3.8a – 5.3         Write the equation.
1.7 + a – a = 3.8a – a – 5.3     Subtract a from each side.
        1.7 = 2.8a – 5.3         Simplify.
  1.7 + 5.3 = 2.8a – 5.3 + 5.3 Add 5.3 to each side.
        7.0 = 2.8a               Simplify.

                                 Divide each side by 2.8.

        2.5 = a                  Check your solution.
Answer: The solution is 2.5
A. Solve 4x + 15 = 2x – 7.

A.


B.
                                      0%


                             1.           A
C. –11
                             2.           B
                             3.           C
D. –22                       4.           D
                                  A   B    C   D
B. Solve 2.4 – 3m = 6.4m – 8.88.

A. 3.3


B. 1.2
                                            0%


                                   1.           A
C. –0.7
                                   2.           B
                                   3.           C
D. –1.9                            4.           D
                                        A   B    C   D
CAR RENTAL A car rental agency has two plans.
Under plan A, a car rents for $80 plus $20 each day.
Under plan B, a car rents for $120 plus $15 each day.
What number of days results in the same cost?
Let d represent the number of days.
$80 plus $20 for each day equals $120 plus $15 for each
day
       80 + 20d = 120 + 15d         Write the original
                                    equation.
80 + 20d – 15d = 120 + 15d – 15d Subtract 15d from
                                    each side.
       80 + 5d = 120                Simplify.
 80 – 80 + 5d = 120 – 80         Subtract 80 from
                                 each side.
          5d = 40                Simplify.

                                 Divide each side by
                                 5.
            d= 8                 Simplify.

Answer: The cost would be the same for 8 days.
CELL PHONES A cell phone provider offers two
plans. Under plan A, the monthly cost is $20 with a
cost of $0.35 per minute. Under plan B, the monthly
cost is $35 with a cost of $0.15 per minute. What
number of minutes results in the same cost?
A. 30 minutes
                                0%
                                             1.   A
B. 75 minutes                                2.   B
                                             3.   C
C. 110 minutes                               4.   D

D. 275 minutes
                            A   B    C   D
Five-Minute Check (over Lesson 8-1)
Main Ideas and Vocabulary
Targeted TEKS
Example 1: Solve Equations with Parentheses
Example 2: Use an Equation to Solve a Problem
Example 3: No Solution
Example 4: All Numbers as Solutions
• Solve equations that involve grouping symbols.
• Identify equations that have no solution or an
  infinite number of solutions.



• null or empty set
• identity
8.2 The student selects and uses appropriate
operations to solve problems and justify solutions.
(A) Select appropriate operations to solve
problems involving rational numbers and justify the
selections. (B) Use appropriate operations to solve
problems involving rational numbers in problem
situations. Also addresses TEKS 8.5(A).
           Solve Equations with Parentheses

A. Solve 3h = 5(h – 2). Check your solution.

    3h = 5(h – 2)            Write the equation.
    3h = 5(h) – 5(2)         Use the Distributive
                             Property.
    3h = 5h – 10             Simplify.
3h – 5h = 5h – 5h – 10       Subtract 5h from each
                             side.
   –2h = –10                 Simplify.

                             Divide each side by –2.
            Solve Equations with Parentheses

     h =5                    Simplify.

Check
    3h = 5(h – 2)            Write the equation.
        ?
   3(5) = 5(5 – 2)           Replace h with 5.
        ?
    15 = 5(3)                Simplify.
    15 = 15                 This statement is true.


Answer: The solution is 5.
           Solve Equations with Parentheses

B. Solve 6(b – 2) = 3(b + 8.5). Check your solution.

    6(b – 2) = 3(b + 8.5)     Write the equation.
    6b – 12 = 3b + 25.5       Use the Distributive
                              Property.
6b – 12 – 3b = 3b – 3b + 25.5 Subtract 3b from each
                              side.
    3b – 12 = 25.5            Simplify.
3b – 12 + 12 = 25.5 + 12      Add 12 to each side.
         3b = 37.5            Simplify.
                              Divide each side by 3.
           Solve Equations with Parentheses

          b = 12.5              Simplify.




Answer: The solution is 12.5.
A. Solve 4t = 7(t – 3). Check your solution.

A. –7


B. 1


C.
                                                   A. A
                                                   B. 0% B
D. 7                                     0%   0%             0%
                                                   C. C

                                     A




                                              B




                                                     C




                                                             D
                                                   D. D
B. Solve 3(p + 5) = 6(p – 2). Check your solution.

A. 9


B. 8


C.
                                                   A. A
                                                   B. 0% B
D.                                       0%   0%             0%
                                                   C. C

                                     A




                                              B




                                                     C




                                                             D
                                                   D. D
            Use an Equation to Solve a Problem

GEOMETRY The perimeter of a rectangle is 36
inches. Find the dimensions if the length is 2 inches
greater than three times the width.
Words      2 times the length + 2 times the width =
           perimeter
Variable   Let w = the width.
         Let 3w + 2 = the length.
                                     w
Equation 2(3w + 2) + 2w = 36

                                              3w + 2
            Use an Equation to Solve a Problem

2(3w + 2) + 2w = 36             Write the equation.
  6w + 4 + 2w = 36              Use the Distributive
                                Property.
        8w + 4 = 36             Simplify.
    8w + 4 – 4 = 36 – 4         Subtract 4 from each side.
            8w = 32             Simplify.
             w =4               Mentally divide each side
                                by 8.
Evaluate 3w + 2 to find the length.
     3(4) + 2 = 12 + 2 or 14 Replace w with 4.
Answer: The width is 4 inches. The length is 14 inches.
GEOMETRY The perimeter of a rectangle is 26 feet.
Find the dimensions if the length is 2 feet less than
twice the width.

A.
                                               0%

B.
                                      1.   A
C. width = 5 ft; length = 8 ft        2.   B
                                      3.   C
                                           A   B    C   D
                                      4.   D
D. width = 11 ft; length = 24 ft
            No Solution

Solve 4x – 0.3 = 4x + 0.9.
     4x – 0.3 = 4x + 0.9        Write the equation.
4x – 4x – 0.3 = 4x – 4x + 0.9   Subtract 4x from each
                                side.
        –0.3 = 0.9              Simplify.
Answer: The sentence –0.3 = 0.9 is never true. So, the
        solution set is Ø.


                                          Interactive Lab:
                                    Solving Two-Step Equations
Solve 16 + 1.3m = –12 + 1.3m.

A. 3.1

                                     0%
B. 21.5
                                1.       A
C. The solution set is Ø.       2.       B
                                3.       C
                                4.       D
D. The solution set is all       A   B    C   D


   numbers.
            All Numbers as Solutions

Solve 3(4x – 2) + 15 = 12x + 9.
3(4x – 2) + 15 = 12x + 9     Write the equation.
 12x – 6 + 15 = 12x + 9      Use the Distributive
                             Property.
      12x + 9 = 12x + 9      Simplify.
  12x + 9 – 9 = 12x + 9 – 9 Subtract 9 from each side.
         12x = 12x           Simplify.
            x =x              Mentally divide each side by
                              12.
Answer: The sentence x = x is always true. The
        solution set is all numbers.
Solve 10a – 9 = 5(2a – 3) + 6.

A. 0


B.


C. The solution set is Ø.
                                               A. A
                                               B. 0% B
D. The solution set is all           0%   0%             0%
                                               C. C
   numbers.
                                 A




                                          B




                                                 C




                                                         D
                                               D. D
Five-Minute Check (over Lesson 8-2)
Main Ideas and Vocabulary
Targeted TEKS
Example 1: Write Inequalities
Key Concept: Inequalities
Example 2: Real-World Example
Example 3: Determine Truth of an Inequality
Example 4: Graph Inequalities
Example 5: Write an Inequality
• Write inequalities.
• Graph inequalities.



• inequality
8.2 The student selects and uses appropriate
operations to solve problems and justify solutions.
(A) Select appropriate operations to solve
problems involving rational numbers and justify the
selections. (B) Use appropriate operations to solve
problems involving rational numbers in problem
situations. Also addresses TEKS 8.4(A).
            Write Inequalities

A. Write an inequality for the following sentence.
Your height is greater than 52 inches.
Words      Your height is greater than 52 inches.

Variable   Let h represent height.


Inequality h > 52

Answer: h > 52
           Write Inequalities

B. Write an inequality for the following sentence.
Your speed is less than or equal to 62 mph.
Words      Your speed is less than or equal to 62 mph.

Variable   Let s represent speed.


Inequality s ≤ 62

Answer: s ≤ 62
A. Write an inequality for the following sentence.
Your height is less than 48 inches.

A. h < 48


B. h > 48


C. h ≤ 48                                         A. A
                                        0%   0%
                                                  B. 0% B   0%
                                                  C. C
D. h ≥ 48
                                    A




                                             B




                                                    C




                                                            D
                                                  D. D
B. Write an inequality for the following sentence.
Your age is greater than 12 years.

A. a < 12


B. a > 12


C. a ≤ 12                                         A. A
                                        0%   0%
                                                  B. 0% B   0%
                                                  C. C
D. a ≥ 12
                                    A




                                             B




                                                    C




                                                            D
                                                  D. D
ENVIRONMENT To meet a certain air quality
standard, an automobile must have a fuel efficiency
of not less than 27.5 miles per gallon. Write an
inequality to describe this situation.

Words      Fuel efficiency of at least 27.5 miles per gallon

Variable   Let e = fuel efficiency in miles per gallon


Inequality e ≥ 27.5

Answer: The inequality is e ≥ 27.5.
CAFETERIA The school cafeteria allows each
student no more than 2 servings of dessert during
lunch. Write an inequality to describe this situation.

A. s < 2

                                                0%

B. s > 2
                                       1.           A
                                       2.           B
C. s ≤ 2                               3.           C
                                       4.           D
                                            A   B    C   D

D. s ≥ 2
           Determine Truth of an Inequality

A. For the given value, state whether the inequality
is true or false.
s – 9 < 4, s = 6

s–9 < 4                  Write the inequality.
     ?
6–9<4                    Replace s with 6.

  –3 < 4                 Simplify.

Answer: The sentence is true.
              Determine Truth of an Inequality

B. For the given value, state whether the inequality
is true or false.



                           Write the inequality.

                           Replace a with 36.
   ?
14 ≤ 12 + 1                Simplify.
14 ≤ 13                    Simplify.

Answer: The sentence is false.
A. For the given value, state whether the inequality is
true or false.
12 – m > 7, m = 5

A. true                                       0%




B. false                                 1.       A
                                         2.       B
                                         3.       C
C. sometime true                         4.       D
                                          A   B    C   D




D. cannot be determined
B. For the given value, state whether the inequality is
true or false.


                                              0%
A. true
                                         1.       A
B. false                                 2.       B
                                         3.       C
                                         4.       D
C. sometime true                          A   B    C   D




D. cannot be determined
           Graph Inequalities

A. Graph x > 10 on a number line.




Answer:


The open circle means the number 10 is not included in
the graph.
           Graph Inequalities

B. Graph x ≥ 10 on a number line.




Answer:


The closed circle means the number 10 is included in
the graph.
           Graph Inequalities

C. Graph x < 10 on a number line.




Answer:


The open circle means the number 10 is not included in
the graph.
           Graph Inequalities

D. Graph x ≤ 10 on a number line.




Answer:


The closed circle means the number 10 is included in
the graph.
A. Graph x < 3 on a number line.

A.


B.


C.
                                                 A. A
                                                 B. 0% B
D.                                     0%   0%             0%
                                                 C. C

                                   A




                                            B




                                                   C




                                                           D
                                                 D. D
B. Graph x > 3 on a number line.

A.


B.


C.
                                                 A. A
                                                 B. 0% B
D.                                     0%   0%             0%
                                                 C. C

                                   A




                                            B




                                                   C




                                                           D
                                                 D. D
C. Graph x ≤ 3 on a number line.

A.


B.


C.
                                                 A. A
                                                 B. 0% B
D.                                     0%   0%             0%
                                                 C. C

                                   A




                                            B




                                                   C




                                                           D
                                                 D. D
D. Graph x ≥ 3 on a number line.

A.


B.


C.
                                                 A. A
                                                 B. 0% B
D.                                     0%   0%             0%
                                                 C. C

                                   A




                                            B




                                                   C




                                                           D
                                                 D. D
            Write an Inequality

Write the inequality for the graph.




A closed circle is on –38, so the point –38 is included in
the graph. The arrow points to the right, so the graph
includes all numbers greater than or equal to –38.


Answer: The inequality is x ≥ –38.
Write the inequality for the graph.




A. x < –7


B. x > –7
                                                    A. A
                                                    B. 0% B
C. x ≤ –7                                 0%   0%             0%
                                                    C. C

                                      A




                                               B




                                                      C




                                                              D
                                                    D. D
D. x ≥ –7
Five-Minute Check (over Lesson 8-3)
Main Idea
Targeted TEKS
Key Concept: Addition and Subtraction Properties
Example 1: Solve an Inequality Using Subtraction
Example 2: Solve an Inequality Using Addition
Example 3: Graph Solutions of Inequalities
Example 4: Real-World Example
• Solve inequalities by using the Addition and
  Subtraction Properties of Inequality.
8.2 The student selects and uses appropriate
operations to solve problems and justify solutions.
(A) Select appropriate operations to solve
problems involving rational numbers and justify the
selections. (B) Use appropriate operations to solve
problems involving rational numbers in problem
situations. Also addresses TEKS 8.4(A).
             Solve an Inequality Using Subtraction

Solve y + 5 > 11. Check your solution.
   y + 5 > 11             Write the inequality.
y + 5 – 5 > 11 – 5        Subtract 5 from each side.
        y>6               Simplify.
To check your solution, try any number greater than 6.
Check
   y + 5 > 11             Write the inequality.
         ?
   7 + 5 > 11             Replace y with 7.
      12 > 11            This statement is true.
Answer: The solution is y > 6.
Solve x + 9 < 13.

A. x < 22


B. x < 4


C. x < –4
                                  A. A
                                  B. 0% B
D. x > 4                0%   0%             0%
                                  C. C

                    A




                             B




                                    C




                                            D
                                  D. D
           Solve an Inequality Using Addition

Solve –21 ≥ d – 8. Check your solution.
    –21 ≥ d – 8          Write the inequality.
–21 + 8 ≥ d – 8 + 8      Add 8 to each side.
    –13 ≥ d              Simplify.




Answer: The solution is –13 ≥ d or d ≤ –13.
Solve m + 8 < –2. Check your solution.

A. m > 10


B. m < 6
                                              0%


                                     1.           A
C. m < –6
                                     2.           B
                                     3.           C
D. m < –10                           4.           D
                                          A   B    C   D
        Graph Solutions of Inequalities

Solve       Graph the solution on a number line.

                  Write the inequality.
           Graph Solutions of Inequalities

                      Simplify.

Answer:


Graph the solution.

Answer:
Solve   Graph the solution on a number line.


A.
                                        0%



B.
                               1.   A
C.                             2.   B
                               3.   C
                                    A   B    C   D




D.                             4.   D
BOWLING Katya took $12 to the bowling alley. Shoe
rental costs $3.75. What is the most she can spend on
games and snacks?

Explore      We need to find the greatest amount Katya
             can spend on games and snacks.
Plan         Let x represent the amount Katya can
             spend on games and snacks. Write an
             inequality to represent the problem. Recall
             that at most means less than or equal to.
Words      Cost of shoes plus cost of games and snacks
           must be less than or equal to total.
Variable   Let x equal the cost of games and snacks.


Inequality 3.75 + x ≤ 12

Solve      3.75 + x ≤ 12          Write the inequality.
    3.75 – 3.75 + x ≤ 12 – 3.75   Subtract 3.75 from
                                  each side.
                  x ≤ 8.25        Simplify.
Check    Check by choosing an amount less than or
         equal to $8.25, say, $6. Then Katya would
         spend $3.75 + $6 or $9.75 in all. Since $9.75
         < $12, the answer is reasonable.



Answer: The most Katya can spend on games and
        snacks is $8.25.
MOVIES Danielle has $10 to take to the movies. If the
cost of a ticket is $4.50, what is the most she can
spend on snacks?

A. $5.50


B. $6.00

                                                 A. A
C. $6.50
                                       0%   0%
                                                 B. 0% B   0%
                                                 C. C

                                   A




                                            B




                                                   C




                                                           D
D. $14.50                                        D. D
Five-Minute Check (over Lesson 8-4)
Main Ideas
Targeted TEKS
Key Concept: Multiplication and Division Properties
Example 1: Multiply or Divide by a Positive Number
Example 2: Test Example
Key Concept: Multiplication and Division Properties
Example 3: Multiply or Divide by a Negative Number
• Solve inequalities by multiplying and dividing by a
  positive number.
• Solve inequalities by multiplying or dividing by a
  negative number.
8.2 The student selects and uses appropriate
operations to solve problems and justify solutions.
(A) Select appropriate operations to solve
problems involving rational numbers and justify the
selections. (B) Use appropriate operations to solve
problems involving rational numbers in problem
situations. Also addresses TEKS 8.4(A).
           Multiply or Divide by a Positive Number

A. Solve 9x ≤ 54. Check your solution.

9x ≤ 54                   Write the inequality.

                         Divide each side by 9.

 x≤6                      Simplify.


Answer: The solution is x ≤ 6. You can check this
        solution by substituting 6 or a number less
        than 6 into the inequality.
            Multiply or Divide by a Positive Number

B. Solve        Check your solution.

                         Write the inequality.


                         Multiply each side by 9.

   d > 36                Simplify.

Answer: The solution is d > 36. You can check this
        solution by substituting 36 or a number greater
        than 36 into the inequality.
A. Solve 3y > 21. Check your solution.

A. y > 63


B. y > 7


C. y < 18
                                                 A. A
                                                 B. 0% B
D. y < 7                               0%   0%             0%
                                                 C. C

                                   A




                                            B




                                                   C




                                                           D
                                                 D. D
B. Solve    Check your solution.

A. p ≥ 2


B. p ≥ 3


C. p ≥ 18
                                             A. A
                                             B. 0% B
D. p ≤ 18                          0%   0%             0%
                                             C. C

                               A




                                        B




                                               C




                                                       D
                                             D. D
Martha earns $9 per hour working at a fast food
restaurant. Which inequality can be used to find how
many hours she must work in a week to earn at least
$117?
A 9x < 117 B 9x ≥ 117 C 9x > 117 D 9x ≤ 117

Read the Test Item
You are to write an inequality to represent a real-world
problem.
Solve the Test Item
Words        Amount earned per hour times number of
             hours is at least amount earned each week.
Variable   Let x represent the number of hours worked.


Inequality 9 ● x ≥ 117




Answer: The answer is B.
TEST EXAMPLE Ed earns $6 per hour working at the
library. Which inequality can be used to find how
many hours he must work in a week to earn more
than $100?
A. 6x < 100
                                           0%

B. 6x ≥ 100
                                  1.           A
                                  2.           B
C. 6x ≤ 100                       3.           C
                                  4.           D
D. 6x > 100                            A   B    C   D
           Multiply or Divide by a Negative Number

A. Solve          and check your solution. Then
graph the solution on a number line.

                         Write the inequality.


                         Multiply each side by –5 and
                         reverse the symbol.

Answer: x ≤ –35          Check this result.
Graph the solution x ≤ –35.
            Multiply or Divide by a Negative Number

B. Solve –9x < –27 and check your solution. Then
graph the solution on a number line.
–9x < –27                Write the inequality.

                         Divide each side by –9 and
                         reverse the symbol.


Answer: x > 3            Check this result.
Graph the solution, x > 3.
A. Solve       and check your solution. Then graph
your solution on a number line.
A. x < 18
                                              0%




B. x < –2
                                     1.       A
                                     2.       B
C. x < –18                           3.
                                          A
                                              C
                                              B    C   D

                                     4.       D

D. x > –18
B. Solve –5x ≤ –40 and check your solution. Then
graph your solution on a number line.

A. x ≥ –45
                                               0%




B. x ≥ –8
                                      1.       A
                                      2.       B
C. x ≤ 8                              3.
                                           A
                                               C
                                               B    C   D

                                      4.       D

D. x ≥ 8
Five-Minute Check (over Lesson 8-5)
Main Idea
Targeted TEKS
Example 1: Solve a Two-Step Inequality
Example 2: Reverse the Inequality Symbol
Example 3: Real-World Example
• Solve inequalities that involve more than one
  operation.
8.2 The student selects and uses appropriate
operations to solve problems and justify solutions.
(A) Select appropriate operations to solve
problems involving rational numbers and justify the
selections. (B) Use appropriate operations to solve
problems involving rational numbers in problem
situations. Also addresses TEKS 8.4(A).
           Solve a Two-Step Inequality

Solve 5x + 13 > 83 and check your solution. Graph
the solution on a number line.
    5x + 13 > 83         Write the inequality.
5x + 13 – 13 > 83 – 13   Subtract 13 from each side.
         5x > 70         Simplify.
                         Divide each side by 5.

Answer: x > 14           Simplify.

                                         BrainPOP:
                                     Solving Inequalities
            Solve a Two-Step Inequality

Check 5x + 13 > 18            Write the inequality.
                ?
     5(15) + 13 > 83          Replace x with a number
                              greater than 14. Try 15.
                ?
        75 + 13 > 83          Simplify.
             88 > 83         The solution checks.

Graph the solution, x > 14.
Answer: x > 14
Solve 3x – 9 < 18 and check your solution. Graph the
solution on a number line.

A. x < 3


B. x < 9


C. x > 3                                         A. A
                                       0%   0%
                                                 B. 0% B   0%
                                                 C. C
D. x > 9
                                   A




                                            B




                                                   C




                                                           D
                                                 D. D
           Reverse the Inequality Symbol

Solve 7 – 4a ≤ 23 – 2a and check your solution.
Graph the solution.
    7 – 4a ≤ 23 – 2a            Write the inequality.
7 – 4a + 2a ≤ 23 – 2a + 2a       Add 2a to each side.
    7 – 2a ≤ 23                  Simplify.
 7 – 7 – 2a ≤ 23 – 7             Subtract 7 from each
                                 side.
       –2a ≤ 16                  Simplify.

                                 Divide each side by –2
                                 and change ≤ to ≥.
Answer: a ≥ –8                   Simplify.
            Reverse the Inequality Symbol

Check your solution by substituting a number greater
than or equal to –8.

Graph the solution, a ≥ –8.

Answer:
Solve 8 + 2x < 5x – 7 and check your solution. Graph
the solution on a number line.

A. x > 5

                                              0%

B. x < 5
                                     1.   A
                                     2.   B
C.
                                     3.   C
                                          A   B    C   D

                                     4.   D
D.
RUNNING José wants to run a 10k marathon. Refer to
Get Ready for the Lesson. If the length of his current
daily run is 2 kilometers, how many kilometers should
he increase his daily run to have enough endurance
for the race?

Words      3 times 2 plus amount of increase is greater
           than or equal to desired distance
Variable   Let d = the amount of increase.


Inequality 3  (2 + d) ≥ 10
  3(2 + d) ≥ 10       Write the inequality.

   6 + 3d ≥ 10        Use the Distributive
                      Property.
6 + 3d – 6 ≥ 10 – 6   Subtract 6 from each
                      side.
       3d ≥ 4         Simplify.

                      Divide each side by 3.


                      Simplify.
Answer: At least   km
BACKPACKING A person weighing 168 pounds has
a 7-pound backpack. If three times the weight of
your backpack and its contents should be less than
your body weight, what is the maximum weight for
the contents of the pack?
                                           0%


A. greater than 49 pounds
                                      1.       A
B. less than 49 pounds                2.       B
                                      3.       C
C.                                    4.       D
                                       A   B    C   D



D. less than 147 pounds
Five-Minute Checks

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Solving Two-Step Equations

Two-Step Equations

Solving Inequalities
Lesson 8-1 (over Chapter 7)
Lesson 8-2 (over Lesson 8-1)
Lesson 8-3 (over Lesson 8-2)
Lesson 8-4 (over Lesson 8-3)
Lesson 8-5 (over Lesson 8-4)
Lesson 8-6 (over Lesson 8-5)
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                           (over Chapter 7)


Which of the following options correctly states and
explains whether the given relation is a function?
{(–1, 2), (0, 1), (2, 1), (0, 2)}

A. No; 0 is paired with
   1 and 2.
B. No; –1 is paired with
   –2 and 0.
                                                  A. A
C. Yes; there is no many-to-
   one relation present.                0%   0%
                                                  B. 0% B   0%
                                                  C. C
D. Yes; there is no one-to-
                                    A




                                             B




                                                    C




                                                            D
                                                  D. D
   many relation present.
                           (over Chapter 7)


Find the x-intercept and y-intercept for the graph of
2x + y = 6.

A. 2, 1

                                               0%
B. 3, 1
                                      1.           A
                                      2.           B
C. 2, 6                               3.           C
                                      4.           D
                                           A   B    C   D
D. 3, 6
                           (over Chapter 7)


Find the slope of the line that passes through the
pair of points A(4, 2) and B(–4, –4).

A.
                                             0%


B.                                          1.        A
                                            2.        B
                                            3.        C
C. 0
                                            4.        D

                                        A    B    C   D
D. undefined
                           (over Chapter 7)


State the slope and the y-intercept for the graph of
the equation x + 2y = –8.

A.


B.


C. –2, –4                                         A. A
                                        0%   0%
                                                  B. 0% B   0%
                                                  C. C
D. –4, –2
                                    A




                                             B




                                                    C




                                                            D
                                                  D. D
                           (over Chapter 7)


Which ordered pair is not a solution of

A. (1, –1)


B. (–2, –5)                                    0%


                                      1.           A
C. (–2, –3)                           2.           B
                                      3.           C
                                      4.           D
D. (–1, –1)                                A   B    C   D
                      (over Lesson 8-1)


Solve 2x + 15 = 7x.

A. –3


B.


C.
                                             A. A
                                   0%   0%
                                             B. 0% B   0%
D. 3                                         C. C

                               A




                                        B




                                               C




                                                       D
                                             D. D
                      (over Lesson 8-1)


Solve 5a = 3a – 18.

A. –9


B.                                        0%


                                 1.           A
C.                               2.           B
                                 3.           C
                                 4.           D
D.
                                      A   B    C   D
                            (over Lesson 8-1)


Solve 1.3 – 3b = b – 0.3.

A. 1.4

                                              0%
B. 1.04
                                             1.        A
                                             2.        B
C. 0.4
                                             3.        C
                                             4.        D
D. 0.14
                                         A    B    C   D
                           (over Lesson 8-1)


Solve 4x – 7 = –5x – 25.

A. –18


B. –2


C.
                                                  A. A
                                        0%   0%
                                                  B. 0% B   0%
D.                                                C. C

                                    A




                                             B




                                                    C




                                                            D
                                                  D. D
                            (over Lesson 8-1)


Solve 7.5 + n = 2.7n – 1.

A. 5


B. 4                                            0%


                                       1.           A
C. –4.8                                2.           B
                                       3.           C
                                       4.           D
D. –5.8
                                            A   B    C   D
                           (over Lesson 8-1)



What is the solution of 6 – 2a = 2a – 6?

A. –3
                                             0%

B. –1
                                            1.        A
                                            2.        B
C. 0                                        3.        C
                                            4.        D

D. 3                                    A    B    C   D
                      (over Lesson 8-2)


Solve 2(x – 5) = 8.

A. 13


B. 9


C. 6.5
                                             A. A
                                   0%   0%
                                             B. 0% B   0%
D. 4.5                                       C. C

                               A




                                        B




                                               C




                                                       D
                                             D. D
                        (over Lesson 8-2)


Solve 3(2a – 1) = 42.

A. 6.8


B. 7.1                                      0%


                                   1.           A
C. 7.3                             2.           B
                                   3.           C
                                   4.           D
D. 7.5
                                        A   B    C   D
                             (over Lesson 8-2)


Solve 6(n – 4) = 3(n + 3).

A. 11

                                               0%
B. 2.3
                                              1.        A
                                              2.        B
C. –2.5
                                              3.        C
                                              4.        D
D. –5
                                          A    B    C   D
                               (over Lesson 8-2)


Solve 5(x – 4) = 2(x – 2.5).

A. 1


B. 2.1


C. 5
                                                      A. A
                                            0%   0%
                                                      B. 0% B   0%
D. 8.3                                                C. C

                                        A




                                                 B




                                                        C




                                                                D
                                                      D. D
                           (over Lesson 8-2)


Find the dimensions of the
rectangle shown in the figure.
The perimeter is 90 meters.

A. w = 8 m; ℓ = 27 m
                                               0%


B. w = 9 m; ℓ = 31 m                  1.           A
                                      2.           B
C. w = 19 m; ℓ = 71 m                 3.           C
                                      4.           D
                                           A   B    C   D

D. w = 10 m; ℓ = 35 m
                            (over Lesson 8-2)



Which equation has no solution?

A. 12n – 6 = 2(n + 1)
                                              0%

B. 8n – 7 = 5(n – 5)
                                             1.        A
                                             2.        B
C. 5(n + 4) + 8 = 5n + 20                    3.        C
                                             4.        D

D. 4(2n + 6) = 4(n + 3)                  A    B    C   D
                          (over Lesson 8-3)


Write an inequality for the sentence. A number
decreased by 7 is at most 9.

A. n – 7 > 9


B. n – 7 < 9


C.                                               A. A
                                       0%   0%
                                                 B. 0% B   0%
                                                 C. C
D.
                                   A




                                            B




                                                   C




                                                           D
                                                 D. D
                         (over Lesson 8-3)


Write an inequality for the sentence. There are
more than 500 students (s) at Candlewood Middle
School.

A. s > 500
                                             0%


B. s < 500                          1.           A
                                    2.           B
C.                                  3.           C
                                    4.           D
                                         A   B    C   D

D.
                            (over Lesson 8-3)


For n = 4, state whether the inequality 15 – n < 9 is
true or false.

A. true
                                                0%
                                                0%
B. false



                                           A    B


                                                     1.   A
                                                     2.   B
                            (over Lesson 8-3)


For a = 7, state whether the inequality 6a ≥ 42 is
true or false.

A. true
                                                0%
                                                0%
B. false



                                           A    B


                                                     1.   A
                                                     2.   B
                           (over Lesson 8-3)


Write the inequality for
the graph.

A. x < –2

                                               0%
B.
                                      1.           A
                                      2.           B
C. x > –2                             3.           C
                                      4.           D
                                           A   B    C   D
D.
                           (over Lesson 8-3)



If Mike collects 3 more model planes, he will have
at least 10 model planes. Which inequality
represents this situation?

A. p + 3 > 10                                0%



                                            1.        A
B. p + 3 < 10                               2.        B
                                            3.        C
C.                                          4.        D

                                        A    B    C   D


D.
                            (over Lesson 8-4)


Solve the inequality k + 5 > –2.

A. k > –7


B. k < –7


C. k > 7
                                                   A. A
                                         0%   0%
                                                   B. 0% B   0%
D. k < 7                                           C. C

                                     A




                                              B




                                                     C




                                                             D
                                                   D. D
                       (over Lesson 8-4)


Solve the inequality

A.


B.                                         0%


                                  1.           A
C.                                2.           B
                                  3.           C
                                  4.           D
D.
                                       A   B    C   D
                       (over Lesson 8-4)


Solve the inequality

A.

                                         0%
B.
                                        1.        A
                                        2.        B
C.
                                        3.        C
                                        4.        D
D.
                                    A    B    C   D
     (over Lesson 8-4)




A.


B.


C.
                            A. A
                  0%   0%
                            B. 0% B   0%
D.                          C. C

              A




                       B




                              C




                                      D
                            D. D
                            (over Lesson 8-4)


Solve the inequality –8 < x – (–4).

A. x > 12


B. x < 12                                       0%


                                       1.           A
C. x < –12                             2.           B
                                       3.           C
                                       4.           D
D. x > –12
                                            A   B    C   D
                          (over Lesson 8-4)



Tina can spend $20 at most on 2 birthday presents.
If she spends $9.50 on one present, which
inequality represents all the possible amounts p
that she can spend on other presents?

A.
                               0%
                                            1.   A
B.                                          2.   B
                                            3.   C
C.                                          4.   D


D.
                           A   B    C   D
                           (over Lesson 8-5)


Solve the inequality 2x > 22.

A. x > 11


B. x > 20


C. x < 11
                                                  A. A
                                        0%   0%
                                                  B. 0% B   0%
D. x < 20                                         C. C

                                    A




                                             B




                                                    C




                                                            D
                                                  D. D
                       (over Lesson 8-5)


Solve the inequality

A.


B.                                         0%


                                  1.           A
C.                                2.           B
                                  3.           C
                                  4.           D
D.
                                       A   B    C   D
                       (over Lesson 8-5)


Solve the inequality

A.

                                         0%
B.
                                        1.        A
                                        2.        B
C.
                                        3.        C
                                        4.        D
D.
                                    A    B    C   D
                            (over Lesson 8-5)


Solve the inequality 0.8r < –24.

A. r < –30


B. r > –30


C. r < –0.3
                                                   A. A
                                         0%   0%
                                                   B. 0% B   0%
D. r > –0.3                                        C. C

                                     A




                                              B




                                                     C




                                                             D
                                                   D. D
     (over Lesson 8-5)




A.


B.                       0%


                1.           A
C.              2.           B
                3.           C
                4.           D
D.
                     A   B    C   D
                           (over Lesson 8-5)



The product of an integer and –9 is less than –45.
Find the least integer that meets this condition.

A. –6
                                             0%


B. –5                                       1.        A
                                            2.        B
                                            3.        C
C. 5
                                            4.        D

D. 6                                    A    B    C   D
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