Grade 7 Workbook — Chapter 11: One- and Two-Step Inequalities

SOL 7.PFA.4 · Companion to Textbook Chapter 11

Each page below is one Canva page. Headings are sized for direct paste: page title as H1, section labels as H2. Figures referenced by filename live in ../figures/. Item numbers match the textbook exactly, so one answer key serves both books. Wherever a graph is asked for, use the blank number lines provided on the page.

PAGE 1 — Chapter opener

Chapter 11 · One- and Two-Step Inequalities

Standard 7.PFA.4

In this chapter you will:

• Solve one- and two-step inequalities in one variable
• Investigate and explain what multiplying or dividing by a negative number does to a solution set
• Graph solution sets on a number line
• Write inequalities for real situations, and write real situations for inequalities
• Decide which numerical values belong to a solution set
• Compare solving inequalities with solving equations

Words to know: inequality · strict inequality · inclusive inequality · solution · solution set · properties of inequality · additive inverse property · multiplicative identity property · reciprocal · reversal rule · boundary value · open circle · closed circle · ray · constraint

Chapter 10 solved two-step equations. Almost every move carries over. Two things change: the answer is usually a whole range of numbers, and multiplying or dividing by a negative number reverses the symbol.

FIGURE: fig2-equation-vs-inequality.png (full width)

PAGE 2 — Four symbols

11.1 What an Inequality Says

Fill in the table.

Fill in the blanks.

The symbols $<$ and $>$ are called ______________ inequalities. The boundary number ______ (is / is not) a solution.

The symbols $\le$ and $\ge$ are called ______________ inequalities. The boundary number ______ (is / is not) a solution.

The wide end of the symbol always opens toward the ______________ amount.

Words into symbols. Write the symbol that matches each phrase.

PAGE 3 — Solution sets and testing values

The Solution Set

FIGURE: fig1-testing-values.png (full width)

Fill in the blanks.

A ______________ of an inequality is a value that makes the sentence true.

The ______________ ______________ is every value that makes it true.

The equation $x = 4$ has ______ solution. The inequality $x \ge 4$ has ______________ many.

To test a value, substitute it into the ______________ inequality, simplify each side, and decide whether the sentence is ______________.

Test each value against $x \ge -4$. Circle true or false.

Guided practice

1. Write in words what $x \le 7$ says. ____________________________________________

   Is $x = 7$ a solution? ______ Why? ____________________________________________

2. Test each value in $x + 4 > 10$.

   $x = 5$: ______ $> 10$ → ______ $x = 6$: ______ $> 10$ → ______ $x = 7$: ______ $> 10$ → ______

3. Which values from $-2$, $0$, $3$, $4$ are solutions of $2x - 1 \le 5$?

   $x = -2$: ______ $x = 0$: ______ $x = 3$: ______ $x = 4$: ______

   Solutions: ____________________

4. A backpack may weigh at most 15 pounds. Let $w$ = ____________________________

   Inequality: ______________ Is a 15-pound backpack allowed? ______

PAGE 4 — Independent practice 11.1

Practice · Lesson 11.1

5. Write in symbols using $n$.

   a) $n$ is greater than $-4$: ______________ b) $n$ is at least 9: ______________

   c) $n$ is no more than 2.5: ______________ d) $n$ is fewer than 30: ______________

6. Solution or not? Show the substitution.

   a) $x = 3$ in $4x + 1 > 13$: ______ $> 13$ → ______

   b) $x = 3$ in $4x + 1 \ge 13$: ______ $\ge 13$ → ______

   c) $x = -5$ in $2x + 3 \ge -7$: ______ $\ge -7$ → ______

   d) $x = \tfrac{1}{2}$ in $6x - 2 < 1$: ______ $< 1$ → ______

7. Which values from $-4$, $-1$, $0$, $2$, $5$ are solutions of $3x + 2 < 8$? ____________________

8. Three solutions of $x \ge -2$, including one that is not an integer and one that is negative:

   ______, ______, ______

9. Why does $x < 6$ have infinitely many solutions while $x = 6$ has exactly one?

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10. Application. A roller coaster requires riders to be at least 48 inches tall.

    Let $h$ = ____________________ Inequality: ______________

    May a rider exactly 48 inches tall ride? ______ Why? ____________________________

11. Reasoning. Name the one number that is a solution of $x \ge 5$ but not of $x > 5$: ______

    Explain: _______________________________________________

12. Error analysis. Miguel says $x = 4$ solves $5x - 3 < 17$ because $5(4) - 3 = 17$.

    His mistake: _______________________________________________

    Is 4 a solution? ______ An inequality with the same left side for which 4 is a solution: ______________

PAGE 5 — Exit ticket 11.1

Exit Ticket · Lesson 11.1

Name: ________________________ Date: ____________

13. "A class has no more than 28 students." Inequality using $s$: ______________

14. Which values from $-3$, $1$, $2$, $6$ are solutions of $x + 4 \ge 6$? ____________________

15. Is $x = -2$ a solution of $-3x < 6$? Substitution: ______ $< 6$ → ______

16. What is the solution set of an inequality, and why is it usually described rather than listed?

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PAGE 6 — The properties of inequality

11.2 Solving One-Step Inequalities

Complete each property.

Addition property. Add the same number to both sides and the comparison stays true in the ______________ direction.

Subtraction property. Subtract the same number from both sides and the direction ______________.

Multiplication property. Multiply both sides by the same ______________ number and the direction stays the same.

Division property. Divide both sides by the same ______________ number and the direction stays the same.

Why the moves are safe.

Adding the same number ______________ both points the same distance along the number line, so which one is on the left ______________ change.

Multiplying by a positive number stretches or shrinks the distance from zero but never moves a point ______________ zero.

The two-sided check. After solving, substitute one value from ______________ the solution set (it should be ______________) and one value from ______________ it (it should be ______________).

Guided practice

17. $x + 9 > 15$

    Property used: ____________________ Solution: ______________

    Inside test: $x =$ ______ → ______ $> 15$ → ______ Outside test: $x =$ ______ → ______ $> 15$ → ______

18. $y - 3 \le 4$ Solution: ______________ Boundary test: ______ $\le 4$ → ______

19. $6x < 42$ Solution: ______________ Inside test: ______ Outside test: ______

20. $\dfrac{x}{5} \ge 3$ Solution: ______________ Boundary test: ______ $\ge 3$ → ______

PAGE 7 — Independent practice 11.2

Practice · Lesson 11.2

21. Solve.

    a) $x + 12 \le 20$ → ______________ b) $m - 7 > -2$ → ______________

    c) $x + 3.5 < 9$ → ______________ d) $w - \tfrac{1}{4} \ge \tfrac{1}{2}$ → ______________

22. Solve.

    a) $8x \ge 56$ → ______________ b) $\dfrac{n}{4} < 6$ → ______________

    c) $0.5x > 7$ → ______________ d) $\tfrac{3}{4}x \le 9$ → ______________

23. $12 > x + 4$ → ______________ Rewritten with the variable first: ______________

24. $x + 5 \le 5$ → ______________ Three solutions, one negative: ______, ______, ______

25. $2x \ge -10$ → ______________ Inside test: ______ Outside test: ______

26. Application. A van carries at most 1{,}500 pounds. Each crate weighs 60 pounds.

    Let $c$ = ____________________ Inequality: ______________ Solution: ______________

    Crates are whole numbers, so the van can carry at most ______ crates.

27. Reasoning. How is the first move in $x + 7 < 12$ the same as in $x + 7 = 12$, and how do the answers differ?

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28. Error analysis. Nadia solves $\dfrac{x}{2} > 6$ by dividing by 2 and writes $x > 3$.

    Her mistake: ____________________________ Correct solution: ______________

    Test $x = 8$: ______ $> 6$ → ______ What this shows: ____________________________

PAGE 8 — Exit ticket 11.2

Exit Ticket · Lesson 11.2

Name: ________________________ Date: ____________

29. $x - 6 \ge 2$ → ______________ Boundary test: ______ $\ge 2$ → ______

30. $7x < 49$ → ______________

31. $\dfrac{x}{3} \le -4$ → ______________

32. Property used to solve $x + 4 > 9$: ____________________ Solution: ______________

PAGE 9 — The experiment

11.3 The Reversal Rule

FIGURE: fig3-reversal-rule.png (full width)

Run the experiment.

Start with the true statement $2 < 6$.

Multiply both sides by $-1$: the left becomes ______ and the right becomes ______.

On the number line, $-2$ is to the ______________ of $-6$, so $-2$ is the ______________ number.

The true statement is $-2$ ______ $-6$. The direction ______________.

Multiplying by $-1$ ______________ every point across zero, and a reflection turns left into ______________.

Complete the table.

PAGE 10 — Sliding is not reflecting

Add a Negative, or Multiply by One?

FIGURE: fig4-shift-versus-flip.png (full width)

Fill in the blanks.

Start with $2 < 6$ and add $-10$ to both sides: ______ ______ ______.

Both points slid ______ units to the left, so the direction ______________ (did / did not) change.

Adding a negative ______________ the points. Multiplying by a negative ______________ them across zero.

The reversal rule. Reverse the direction of the symbol only when you ______________ or ______________ both sides by a ______________ number. In every other case the direction ______________.

Reverses or not? Circle one.

Guided practice

33. $3 < 7$, multiplied by $-1$: ______ ______ ______

34. $-2 < 5$, divided by $-1$: ______ ______ ______

35. $-2x < 14$ → ______________ Test $x = 0$: ______ $< 14$ → ______ Test $x = -7$: ______ $< 14$ → ______

36. $-\dfrac{x}{4} \ge 3$ → ______________ Boundary test: ______ Outside test: ______

PAGE 11 — Independent practice 11.3

Practice · Lesson 11.3

37. Begin with $8 > 5$.

    a) Add $-10$ to both sides: ______ ______ ______

    b) Multiply both sides by $-10$: ______ ______ ______

    c) Which reversed, and why? ____________________________________________

38. Solve.

    a) $-3x \le 21$ → ______________ b) $-6n > -42$ → ______________

    c) $\dfrac{x}{-2} \ge 5$ → ______________ d) $-0.5x < 4$ → ______________

39. Solve. Watch which ones reverse.

    a) $x - 9 > -4$ → ______________ b) $-9x > -4$ → ______________

    c) $x + (-6) \le -1$ → ______________ d) $-\tfrac{2}{3}x \le 8$ → ______________

40. $-x < 3$ → ______________ Inside test: ______ Outside test: ______

41. Reverses or not, and why?

    a) subtract 8 from both sides: ______ because ____________________

    b) divide both sides by $-4$: ______ because ____________________

    c) add $-8$ to both sides: ______ because ____________________

    d) multiply both sides by $\tfrac{1}{2}$: ______ because ____________________

42. Application. A submarine descends 3 meters per second from the surface, so its elevation after $t$ seconds is $-3t$ meters.

    Inequality for "elevation below $-45$ meters": ______________

    Solution: ______________ In a sentence: ____________________________________________

43. Reasoning. $-4 < 6$, multiplied by $-\tfrac{1}{2}$: ______ ______ ______

    Why the direction had to reverse: _______________________________________________

44. Error analysis. Cleo solves $-5x \ge 30$ and writes $x \ge -6$.

    Test $x = -10$: ______ $\ge 30$ → ______ Test $x = 0$: ______ $\ge 30$ → ______

    What the two tests show: ____________________________ Correct solution: ______________

PAGE 12 — Exit ticket 11.3

Exit Ticket · Lesson 11.3

Name: ________________________ Date: ____________

45. $-7x < 42$ → ______________

46. $\dfrac{x}{-3} \ge -2$ → ______________

47. $10 > 2$, multiplied by $-3$: ______ ______ ______

48. Why does subtracting a negative not reverse the symbol while dividing by a negative does?

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PAGE 13 — Two steps, one question

11.4 Solving Two-Step Inequalities

FIGURE: fig5-two-step-solution.png (full width)

The procedure.

First, name the ______________ and the ______________ term, signs included.

Then undo the ______________ or ______________ on both sides. The direction ______________ changes at this step.

Then undo the ______________ or ______________. Ask: is that number ______________? If yes, ______________ the symbol.

Finally, check with a value from ______________ the solution set and a value from ______________ it.

Where the boundary comes from. The boundary value of the solution set is the solution of the matching ______________.

Guided practice

49. $2x + 3 < 11$

    Step 1: $2x <$ ______ (property: ____________________)

    Step 2: $x <$ ______ (property: ____________________)

    Test $x = 0$: ______ $< 11$ → ______ Test $x = 4$: ______ $< 11$ → ______

50. $5x - 4 \ge 16$ → $5x \ge$ ______ → ______________

    Boundary test: ______ $\ge 16$ → ______ Outside test: ______

51. $-3x + 2 \le 14$ → $-3x \le$ ______ → ______________

    The direction reverses at step ______ because ____________________

    Boundary test: ______ $\le 14$ → ______

52. $\dfrac{x}{2} + 6 > 4$ → $\dfrac{x}{2} >$ ______ → ______________

    Test $x = 0$: ______ Test $x = -4$: ______

PAGE 14 — Independent practice 11.4

Practice · Lesson 11.4

53. Solve.

    a) $4x + 7 \le 31$ → ______________ b) $6x - 5 > 13$ → ______________

    c) $9 + 2x \ge -3$ → ______________ d) $\dfrac{x}{3} - 2 < 1$ → ______________

54. Solve. Every part here reverses at some step — mark where.

    a) $-4x + 9 > 25$ → ______________ b) $-2x - 5 \le 7$ → ______________

    c) $10 - 3x \ge 1$ → ______________ d) $-\dfrac{x}{2} + 4 < 6$ → ______________

55. $4x + 3 > 10$ → ______________ (exact)

56. $0.5x - 1.5 \le 2$ → ______________

57. $19 \ge 4x - 5$ → ______________ Variable written first: ______________

58. Application. A gym charges a $20 joining fee plus $12 per month. Marisol can spend at most $100.

    Let $m$ = ____________________ Inequality: ______________ Solution: ______________

    Whole months she can afford: ______ Check: $12($ ______ $) + 20 =$ ______ $\le 100$ → ______

59. Reasoning. $-5x + 1 \ge 26$ → ______________

    The direction reverses at the step where ____________________________________

    It does not reverse at the other step because ____________________________________

60. Error analysis. Theo gets to $-3x < 15$ correctly, then writes $x < -5$.

    Test $x = 0$ in $-3x + 4 < 19$: ______ $< 19$ → ______

    What the test shows: ____________________________ Correct solution: ______________

PAGE 15 — Exit ticket 11.4

Exit Ticket · Lesson 11.4

Name: ________________________ Date: ____________

61. $3x - 8 \le 7$ → ______________

62. $-2x + 6 > 14$ → ______________

63. $\dfrac{x}{4} + 5 \ge 3$ → ______________

64. One way solving a two-step inequality is the same as solving a two-step equation:

    ---

    One way it is different: _______________________________________________

PAGE 16 — Endpoint, circle, direction

11.5 Graphing Solution Sets

FIGURE: fig6-strict-vs-inclusive-endpoint.png (full width)

Fill in the blanks.

Every graph carries three pieces of information: the ______________, the type of ______________, and the ______________ direction.

Use an ______________ circle for $<$ or $>$. It says the boundary number ______ (is / is not) a solution.

Use a ______________ circle for $\le$ or $\ge$. It says the boundary number ______ (is / is not) a solution.

Shade to the ______________ for $>$ or $\ge$. Shade to the ______________ for $<$ or $\le$.

FIGURE: fig7-four-solution-shapes.png (full width)

Complete the summary table.

Variable first. $6 > x$ means the same as ______________. Swap the sides and swap the ______________.

PAGE 17 — Guided practice 11.5

Graph It · Lesson 11.5

Use the number lines below. Mark the endpoint, choose an open or closed circle, and shade with an arrowhead.

FIGURE: fig10-blank-number-lines.png (full width)

65. $x < 4$ Endpoint: ______ Circle: ______________ Shade: ______________

66. $x \ge -2$ Endpoint: ______ Circle: ______________ Shade: ______________

67. $x + 5 \le 9$ → ______________ Endpoint: ______ Circle: ______________ Shade: ______________

FIGURE: fig10-blank-number-lines.png (full width)

68. $6 > x$ rewritten with the variable first: ______________

    Endpoint: ______ Circle: ______________ Shade: ______________

PAGE 18 — Independent practice 11.5

Practice · Lesson 11.5

For every graphing item, state the endpoint, the circle type, and the shading direction, then draw it on a number line.

FIGURE: fig10-blank-number-lines.png (full width)

69. Graph each. a) $x > -5$ b) $x \le 1$ c) $x \ge 0$ d) $x < -1$

    a) endpoint ______, circle ______________, shade ______________

    b) endpoint ______, circle ______________, shade ______________

    c) endpoint ______, circle ______________, shade ______________

    d) endpoint ______, circle ______________, shade ______________

70. Solve and graph. a) $3x < 12$ → ______________ b) $x - 4 \ge -6$ → ______________

71. $-2x \le 10$ → ______________ Endpoint ______, circle ______________, shade ______________

72. $2x - 3 > 5$ → ______________ Endpoint ______, circle ______________, shade ______________

73. A graph shows an open circle at $-3$ shaded to the right.

    Inequality: ______________ Two values in the set: ______, ______ One value not in it: ______

74. Application. A parking garage admits cars no taller than 6.5 feet.

    Let $h$ = ____________________ Inequality: ______________

    Endpoint ______, circle ______________, shade ______________

    Which part of the graph does not describe any real car? ____________________________

75. Reasoning. Why is the graph of $x = 3$ a point but the graph of $x \ge 3$ a ray?

    ---

    What does the mark at 3 show in each? ____________________________________

76. Error analysis. Owen graphs $x \le -2$ with an open circle at $-2$ shaded right.

    Error 1: ____________________ Test value that exposes it: ______

    Error 2: ____________________ Test value that exposes it: ______

    Correct graph: ____________________________________

PAGE 19 — Exit ticket 11.5

Exit Ticket · Lesson 11.5

Name: ________________________ Date: ____________

FIGURE: fig10-blank-number-lines.png (full width)

77. Graph $x > -4$. Endpoint ______, circle ______________, shade ______________

78. $5x + 2 \le 17$ → ______________ Endpoint ______, circle ______________, shade ______________

79. A graph shows a closed circle at 2 shaded to the left. Inequality: ______________

80. When do you use an open circle, when a closed circle, and why?

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PAGE 20 — From words to symbols

11.6 Inequalities in Context

FIGURE: fig9-whole-number-tickets.png (full width)

The path from a situation to an inequality.

1. Read the whole situation.
2. Name the unknown, with ______________.
3. Find the ______________-unit amount and the ______________ amount.
4. Find the limit and choose the ______________.
5. Write, solve, and check against the ______________.

Whole-number constraints. When the variable counts objects that cannot be split, the value must be a ______________ number and cannot be ______________.

Algebra gives $t \le 5.2$ and tacos are whole, so the answer is at most ______ tacos.

Algebra gives $w \ge 8.4$ and weeks are whole, so the answer is at least ______ weeks.

The rounding went in ______________ (the same / different) directions, so test the two whole numbers nearest the ______________ instead of memorizing a rule.

Guided practice

81. Tickets $9 each plus one $6 popcorn; Nia has at most $42.

    Let $t$ = ____________________ Inequality: ______________ Solution: ______________

    Whole tickets: ______

82. "The temperature $t$, in degrees Celsius, stayed above $-5$." Inequality: ______________

83. Rosa has $30, spends $4 per book, and wants at least $10 left.

    Inequality: ______________ Solution: ______________ Whole books: ______

84. Write a situation in context for $2x + 3 \le 15$.

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    Solution: ______________

PAGE 21 — Independent practice 11.6

Practice · Lesson 11.6

85. Parking: $4 entry plus $2 per hour; Jamal can spend no more than $18.

    Inequality: ______________ Solution: ______________

86. Six more than three times a number is at least 27. Inequality: ______________ Solution: ______________

87. Five less than twice a number is less than 9. Inequality: ______________ Solution: ______________

88. Application. An elevator carries at most 1{,}500 pounds. The operator weighs 180 pounds; boxes weigh 55 pounds each.

    Let $b$ = ____________________ Inequality: ______________ Solution: ______________

    Whole boxes: ______

89. Application. Tacos $2.50 each plus a $1.00 packaging fee; Priya has $14.00.

    Inequality: ______________ Solution: ______________ Whole tacos: ______

    How you turned the algebra into a number of tacos: ____________________________________

90. Write a situation in context for $-4x + 50 \ge 10$, where something decreases.

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    Solution: ______________

91. Reasoning. Write a situation in context for $6x + 15 > 51$.

    ---

    Solution: ______________ Practical answer: at least ______

    Why not "at least 6"? ____________________________________

92. Error analysis. Bo writes $12(x + 5) \le 65$ for "shirts $12 each plus a $5 shipping fee, at most $65."

    Why it does not match: ____________________________________

    Correct inequality: ______________ Solution: ______________

    Check the largest whole number: $12($ ______ $) + 5 =$ ______ $\le 65$ → ______

PAGE 22 — Exit ticket 11.6

Exit Ticket · Lesson 11.6

Name: ________________________ Date: ____________

93. Taxi: $3 plus $2 per mile; Lena can spend at most $21. Inequality: ______________ Solution: ______________

94. Four more than five times a number is at most 39. Inequality: ______________ Solution: ______________

95. Write a situation in context for $7x + 10 \le 80$.

    ---

    Solution: ______________

96. When does a context force you to round a fractional answer down to a whole number? Give an example.

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PAGE 23 — Review Parts A and B

Chapter 11 Review · Solving

Part A — Properties of inequality (7.PFA.4a)

97. a) $x + 8 \le 15$ → ______________ b) $m - 6 > -2$ → ______________

    c) $9x \ge 63$ → ______________ d) $\dfrac{x}{5} < -3$ → ______________

98. a) $4x + 5 \le 29$ → ______________ b) $7x - 9 > 12$ → ______________

    c) $\dfrac{x}{2} + 7 \ge 4$ → ______________ d) $\tfrac{3}{4}x - 2 < 7$ → ______________

99. $0.4x + 1.2 \ge 3.6$ → ______________

100. $22 \ge 6x - 8$ → ______________ Variable written first: ______________

101. $5x + 2 < 9$ → ______________ (exact)

Part B — Multiplying or dividing by a negative (7.PFA.4b)

102. Begin with $6 < 9$.

     a) Add $-15$: ______ ______ ______ b) Multiply by $-3$: ______ ______ ______

     c) Which reversed, and why? ____________________________________

103. a) $-8x \le 32$ → ______________ b) $-x > 7$ → ______________

     c) $\dfrac{x}{-5} \ge 2$ → ______________ d) $-\tfrac{2}{3}x < 10$ → ______________

104. $-6x + 5 \ge -19$ → ______________ Boundary test: ______ Outside test: ______

105. Using $2 < 6$ and a number line, explain why multiplying by a negative reverses the symbol and adding a negative does not.

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PAGE 24 — Review Part C

Chapter 11 Review · Graphing (7.PFA.4c)

FIGURE: fig10-blank-number-lines.png (full width)

106. $x \le -1$ Endpoint ______, circle ______________, shade ______________

107. $x > 3.5$ Endpoint ______, circle ______________, shade ______________

108. $4x - 6 \le 6$ → ______________ Endpoint ______, circle ______________, shade ______________

FIGURE: fig10-blank-number-lines.png (full width)

109. $-3x + 1 < 10$ → ______________ Endpoint ______, circle ______________, shade ______________

110. Closed circle at $-4$, shaded right. Inequality: ______________ Two solutions: ______, ______

PAGE 25 — Review Parts D and E

Chapter 11 Review · Writing and Creating

Part D — Writing an inequality from a situation (7.PFA.4d)

111. A checked bag must weigh no more than 50 pounds. Inequality using $w$: ______________

112. A theater seats at most 250; 40 people are seated.

     Inequality for additional people $p$: ______________ Solution: ______________

113. Bike rental: $6 fee plus $4 per hour; Sam can spend at most $30.

     Inequality: ______________ Solution: ______________

Part E — Creating a situation from an inequality (7.PFA.4e)

114. Situation for $3x + 12 \le 45$:

     ---

     Solution: ______________

115. Situation for $10x \ge 250$:

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     Solution: ______________

116. Situation for $-5x + 60 > 20$, where something decreases:

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     Solution: ______________

PAGE 26 — Review Parts F and G

Chapter 11 Review · Context and Solution Sets

Part F — Problems in context (7.PFA.4f)

117. $150 for pizzas at $13 each plus an $8 delivery fee.

     Inequality: ______________ Solution: ______________ Whole pizzas: ______

118. Mia has $62, earns $14 per lawn, needs at least $160.

     Inequality: ______________ Solution: ______________ Whole lawns: ______

119. A 40-gallon tank drains 6 gallons per minute; it holds more than 10 gallons.

     Inequality: ______________ Solution: ______________

Part G — Identifying values in a solution set (7.PFA.4g)

120. Which of $-6$, $-2$, $0$, $3$, $8$ solve $2x + 5 \le 11$?

     $-6$: ______ $-2$: ______ $0$: ______ $3$: ______ $8$: ______ Solutions: ____________________

121. Which of $-5$, $-3$, $0$, $4$ solve $-4x > 12$?

     $-5$: ______ $-3$: ______ $0$: ______ $4$: ______ Solutions: ____________________

122. Three solutions of $3x - 2 \ge 7$, one not an integer: ______, ______, ______

     Show each works: ____________________________________

PAGE 27 — Review Parts H and I

Chapter 11 Review · Comparing and Mixed

Part H — Inequalities versus equations (7.PFA.4h)

123. $4x + 6 = 26$ → ______________ $4x + 6 \le 26$ → ______________

     How they are related: ____________________ How they differ: ____________________

124. Two things in common: ____________________________________

     Two differences: ____________________________________

125. Why does $2x = 8$ have one solution while $2x \le 8$ has infinitely many? How does each graph look?

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Part I — Mixed application and reasoning

FIGURE: fig10-blank-number-lines.png (full width)

126. $-2x + 9 \le 3$ → ______________ Endpoint ______, circle ______________, shade ______________

     Boundary test: ______ $\le 3$ → ______

127. A student says $-3x < 9$ gives $x < -3$. Test $x = 0$: ______ $< 9$ → ______

     What the test shows: ____________________ Correct solution: ______________

128. An inequality graphed as a closed circle at $-2$ shaded right: ______________

     A different two-step inequality with the same solution set: ______________

     Show it: ____________________________________

129. A van holds at most 44 people; 8 chaperones ride; students sit 4 to a bench.

     Inequality for benches $b$: ______________ Solution: ______________ Whole benches: ______

130. The one number separating $x > 4$ from $x \ge 4$: ______

     How the graphs differ: ____________________________________

     A situation calling for $x > 4$: ____________________________________

     A situation calling for $x \ge 4$: ____________________________________