Algebra 1 Workbook — Chapter 9: Linear Inequalities in Two Variables and Their Systems

SOL A.EI.2 (d, e, f, g, h) · Companion to Textbook Chapter 9

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/. Every numbered item is the same problem as in the textbook, so one answer key serves both. Item numbers run continuously from 1 to 118.

PAGE 1 — Chapter opener

Chapter 9 · Linear Inequalities in Two Variables and Their Systems

Standard A.EI.2 (d, e, f, g, h)

In this chapter you will:

• Decide by substitution whether an ordered pair is a solution
• Graph a linear inequality in two variables as a shaded half plane
• Choose a dashed or solid boundary, and shade the correct side with a test point
• Write an inequality — and a system of two — from a real situation
• Graph a system as the overlap of two half planes, and find its corner
• Verify a solution algebraically, graphically, and with technology, and interpret it in context

Words to know: linear inequality in two variables · ordered pair · solution set · half plane · boundary · strict · inclusive · dashed · solid · test point · system of two linear inequalities · overlap · corner · feasible region

Convention: a strict symbol ($<$, $>$) gets a dashed boundary; an inclusive symbol ($\le$, $\ge$) gets a solid one. This is Chapter 3's open circle and closed circle, stretched from a point into a whole line.

PAGE 2 — An interval, and a region

9.1 From an Interval to a Region

FIGURE: fig1-interval-versus-region.png (full width)

Complete the table.

1. A solution of $x > 2$ is _______________________; a solution of $y > x + 1$ is _______________________

   How the two solution sets differ: _______________________________________

2. Is $(-2, 4)$ a solution of $y > x + 1$? Substitution: _______________________ Answer: ______

PAGE 3 — Testing points

Substitution Decides

FIGURE: fig2-testing-points-in-a-half-plane.png (full width)

Frame. Replace $x$ with the ______ coordinate and $y$ with the ______ coordinate, then read whether the sentence is ____________ or ____________.

3. Two solutions: ____________ and ____________ Two non-solutions: ____________ and ____________

   How the picture tells you: _______________________________________

4. Why is $(1,2)$ not a solution, even though it is on the boundary?

   ---

5. Is $(2,5)$ a solution of $y > x + 1$? ______ Substitution: _______________________

6. Is $(4,1)$ a solution of $2x + 3y \le 10$? ______ Substitution: _______________________

7. The pair $(1,5)$ is marked as a solution. Write the substitution that proves it: _______________________

PAGE 4 — Dashed or solid

The Boundary Question

FIGURE: fig3-dashed-versus-solid-boundary.png (full width)

Complete the frame. A strict symbol gives a ____________ boundary, because its points ______ solutions. An inclusive symbol gives a ____________ boundary, because its points ______ solutions. (are / are not)

9. Which panel has $(2,3)$ as a solution? ____________________

   What the two line styles mean: _______________________________________

   The Chapter 3 convention each grew from: _______________________________________

10. When is a boundary dashed? _______________________ When solid? _______________________

PAGE 5 — Practice · is it a solution?

Practice · Substitution

7. Decide for $y \le 3x - 4$.

8. Decide for $x + 4y > 8$.

11. Three solutions of $y < 2x$: ____________ ____________ ____________

    One substitution that confirms: _______________________

PAGE 6 — Reasoning and exit ticket 9.1

Why a Region?

10. Explain. Why can the solution set not be listed the way a solution of a one-variable equation can be?

    ---

11. Find the error. A student says $(0,1)$ solves $y > x + 1$ "because the point is on the line."

    Test it: _______________________ What went wrong? _______________________________

    What would have to change to make the student right? _______________________

12. Apply it. $2x + 3y \ge 20$, with $x$ two-point baskets and $y$ three-point baskets. Is $(4,3)$ a solution? ______

    Substitution: _______________________ What it means about the game: _______________________

Exit Ticket · Lesson 9.1

15. $(-3,0)$ in $y \ge -x - 4$: ______ Substitution: _______________________

16. $(6,2)$ in $y < \tfrac13 x$: ______ Substitution: _______________________

17. Explain. Why is a solution here an ordered pair rather than a single number?

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PAGE 7 — Which side to shade

9.2 Graphing a Linear Inequality

FIGURE: fig4-shade-above-or-below.png (full width)

The four steps.

1. Solve for $\underline{\hspace{2cm}}$, asking the ____________ question at every negative divisor.

2. Graph the ____________, dashed for ______ and ______, solid for ______ and ______.

3. Test a ____________ that is not on the boundary — usually ____________.

4. Shade the side the test point is on if the test was ____________.

5. $y \ge -x + 3$ — shaded region: ____________________ What $(0,0)$ reported: ____________________

6. $y \le -x + 3$ — shaded region: ____________________ What $(0,0)$ reported: ____________________

PAGE 8 — Two graphs, step by step

Practice · The Four Steps

21. $y > 2x - 3$

    Boundary: ____________ Style: ____________ Test point: ______ Result: ______ Shade: ____________

22. $y \le -\tfrac12 x + 4$

    Boundary: ____________ Style: ____________ Test point: ______ Result: ______ Shade: ____________

FIGURE: fig12-blank-grids.png (half page — use Grids A and B)

PAGE 9 — From standard form

Standard Form and the Origin

FIGURE: fig5-standard-form-test-point.png (full width)

Frame. In $Ax + By \le C$, let $y = 0$ to get the ______-intercept and let $x = 0$ to get the ______-intercept. Then test ____________.

23. $3x + 2y \le 12$ $x$-intercept: ____________ $y$-intercept: ____________

    Test-point substitution: _______________________ Shade: ____________________

24. $4x + 5y \ge 20$ $x$-intercept: ____________ $y$-intercept: ____________

    Origin test: _______________________ Shade: ____________________

PAGE 10 — One-variable boundaries

Horizontal and Vertical

FIGURE: fig6-horizontal-and-vertical-boundaries.png (full width)

24. $y < 3$ — boundary: ____________________ Region: ____________________

    $x \ge -2$ — boundary: ____________________ Region: ____________________

25. $x < 4$ — boundary: ____________ Style: ____________ Region: ____________________

    $y \ge -2$ — boundary: ____________ Style: ____________ Region: ____________________

PAGE 11 — Practice · graphing

Practice · Graph Each

25. Complete the table, then graph each on the grids provided.

26. In part b the origin cannot be the test point. Why? _______________________________

    A point that can be used: ____________

FIGURE: fig12-blank-grids.png (full page)

PAGE 12 — The reversal rule, again

When the Symbol Flips

27. Graph $2x - y > 4$.

    Solve for $y$: _______________________ Reversing step: _______________________

    Property that authorizes it: _______________________________

    Boundary: ____________ Style: ____________ Origin test in the original: _______________________

    Shade: ____________________

28. Find the error. A student graphs $y > -2x + 1$ with a solid boundary, shaded below.

    Error 1: _______________________ Error 2: _______________________

    What $(0,0)$ shows: _______________________________________

PAGE 13 — Reading a graph backwards

Write the Inequality

30. Dashed boundary through $(0,2)$ and $(4,4)$, region below shaded.

    Slope: ______ Boundary: ____________ Inequality: ____________ Check with a shaded point: ______________

31. Solid boundary through $(0,-1)$ and $(2,3)$, region above shaded.

    Slope: ______ Boundary: ____________ Inequality: ____________ Check with a shaded point: ______________

32. Explain. Why is the origin the most convenient test point, and when can it not be used?

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PAGE 14 — Application and exit ticket 9.2

Graphing in a Situation

34. Apply it. $\$200$ for supplies: $x$ calculators at $\$25$ and $y$ boxes at $\$20$, so $25x + 20y \le 200$.

    $x$-intercept: ____________ $y$-intercept: ____________

    Why only the first quadrant is drawn: _______________________________________

Exit Ticket · Lesson 9.2

35. $y < -x + 1$ Style: ____________ Shade: ____________________

36. $3x - y \le 6$ Solved for $y$: ____________ Reversal shown: _______________________

    Origin check: _______________________ Shade: ____________________

37. $x \ge 1$ Boundary: ____________ Region: ____________________

38. Explain. The four steps, and which one the reversal rule belongs to.

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PAGE 15 — Words into an inequality

9.3 Writing an Inequality from a Situation

Complete the translation table.

42. "no more than": ______ "at least": ______ "fewer than": ______ "at most": ______

    Boundary style for each: ____________ ____________ ____________ ____________

43. $\$12$ an hour tutoring, $\$9$ an hour at a shop, at least $\$180$: _______________________

44. $x$ is ____________________________________________

    $y$ is ____________________________________________

    Restrictions the situation adds: ____________ and ____________

PAGE 16 — A budget, drawn

The Fair

FIGURE: fig7-budget-inequality-in-context.png (full width)

39. What $5x + 8y \le 80$ says: _______________________________________

    $x$ is ____________________________ $y$ is ____________________________

40. Why is the region a triangle and not a whole half plane?

    ---

    The two extra restrictions: ____________ and ____________

41. Is $(10,6)$ affordable? ______ Arithmetic: _______________________ It means _______________________

PAGE 17 — Practice · writing inequalities

Practice · Build the Model

45. Write each inequality and define both variables.

50. Boundary style for each of 45 a–d, and the phrase that decided it:

    a) ____________ because ____________ b) ____________ because ____________

    c) ____________ because ____________ d) ____________ because ____________

PAGE 18 — The fair, in detail

Apply It · The Fair Budget

46. Apply it. $\$80$; ride bands $\$5$, game passes $\$8$.

    Inequality: _______________________ $x$ is ____________________ $y$ is ____________________

    Two purchases that work: ____________ () ____________ ()

    One that does not: ____________ (____________)

47. Apply it. With exactly $8$ game passes, the greatest number of ride bands is ______

    Work: _______________________________________

48. Both intercepts of $5x + 8y = 80$: ____________ and ____________

    First means _______________________________________

    Second means _______________________________________

PAGE 19 — Reasoning and exit ticket 9.3

Choosing the Symbol

48. Explain. Why do $x \ge 0$ and $y \ge 0$ come with nearly every model here, and what do they do to the picture?

    ---

49. Explain. "at most $\$80$" versus "less than $\$80$": symbol, boundary style, and one purchase they disagree about.

    ---

50. Find the error. A student writes $5 + 8 \le 80$ and calls the purchase affordable.

    What went wrong? _______________________ Correct inequality: _______________________

Exit Ticket · Lesson 9.3

53. At most $10$ hours of practice, piano and guitar: _______________________

    $x$ is ____________________ $y$ is ____________________

54. Cookies $\$4$, pies $\$7$, must raise more than $\$140$: _______________________

55. $x$ is ____________________ $y$ is ____________________ Boundary: ____________ because ____________

56. Apply it. For item 53, is $(6,5)$ possible? ______ Why? _______________________

PAGE 20 — Two shadings at once

9.4 Systems of Two Linear Inequalities

FIGURE: fig8-two-half-planes-together.png (full width)

Frame. A pair is a solution of a system when it satisfies ____________ inequalities. The solution set is the ____________ of the two shadings, not everything either one covers.

57. The two inequalities: ____________________ and ____________________

    Where they overlap: _______________________________________

58. Verify $(0,0)$: first ____________________ second ____________________ Solution? ______

PAGE 21 — The overlap and its corner

The Corner

FIGURE: fig9-overlap-region-and-corner.png (full width)

59. Corner: ____________ Algebra that locates it: _______________________________

    Is the corner a solution? ______ Why? _______________________________

60. Is $(4,-1)$ a solution? ______ Which inequality fails? ____________________

61. Describe the region of $y \le -x + 5$, $y > 2x - 4$ in words:

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PAGE 22 — Practice · graphing systems

Practice · Shade the Overlap

63. Graph each system on the grids provided and complete the table.

64. Corner of 63a, found algebraically: ____________ In the solution set? ______

    How the boundary styles decide: _______________________________________

FIGURE: fig12-blank-grids.png (full page)

PAGE 23 — Practice · membership in a system

Practice · Both, Not Either

67. Decide for $y \le -x + 5$ and $y > 2x - 4$.

65. Explain. Why is the solution set the overlap and not the union?

    ---

66. A system with no solutions: ____________________ and ____________________

    How its graph shows that: _______________________________________

67. Find the error. A student says any point in either shaded region solves the system.

    Use $(4,-1)$ to show why not: _______________________________________

PAGE 24 — A system from a situation

Two Limits at Once

FIGURE: fig10-system-in-context-first-quadrant.png (full width)

62. The system: ____________________ and ____________________

    $x$ is ____________________________ $y$ is ____________________________

63. Apply it. At most $12$ hours; at least $\$96$; $\$8$ an hour babysitting, $\$12$ an hour tutoring.

    System: ____________________ and ____________________ Restrictions: ____________ and ____________

64. Apply it. The three corners: ____________ ____________ ____________

    One of them, interpreted: _______________________________________

PAGE 25 — Applying the system, and exit ticket 9.4

Does This Week Work?

68. Apply it. Five hours babysitting and five hours tutoring.

    Hours check: _______________________ Earnings check: _______________________ Workable? ______

69. Apply it. A schedule that meets the hour limit but misses the earnings goal: ____________

    Arithmetic: _______________________________________

Exit Ticket · Lesson 9.4

73. Graph $y > -2x + 6$ and $y \le x$. Corner: ____________

74. Is $(4,0)$ a solution of that system? ______ Both substitutions: _______________________

75. Explain. How does the graph of a system of two inequalities differ from a system of two equations?

    ---

76. Apply it. At most $\$50$; movie tickets $\$9$, snacks $\$5$; at least $2$ tickets.

    System: ____________________ and ____________________

PAGE 26 — Three checks

9.5 Verifying and Interpreting

FIGURE: fig11-verify-candidates-graphically.png (full width)

Complete the table.

77. A $(4,6)$ — hours: _______________________ earnings: _______________________ Solution? ______

    Where it sits on the graph: _______________________

78. B $(12,0)$ — hours: _______________________ earnings: _______________________ Solution? ______

    What it means that it lies on both boundaries: _______________________________

79. C $(10,1)$ — why it fails: _______________________ Short by: ______

PAGE 27 — Practice · verifying

Practice · Substitute and Check

83. Decide for $3x + 4y \le 24$, and mark boundary points.

84. Decide for the system $y < x + 3$ and $y \ge 2x - 6$.

80. $(2,5)$ in $y \le 3x$ and $y > x - 4$: ______ Substitutions: _______________________

    How the graph confirms it: _______________________________

PAGE 28 — Technology as a check

The Calculator Check

85. Technology. Entering $y \le -x + 5$ and $y > 2x - 4$:

    What you type, line 1: _______________________ line 2: _______________________

    How to read the solution set: _______________________________________

    How to test a candidate point: _______________________________________

86. If the tool's shading disagrees with yours, check in order:

    1. _______________________ 2. _______________________ 3. _______________________

87. Find the error. By hand, $5x - 2y < 10$ was shaded below; the tool shades above.

    Likely error: _______________________ Algebra that settles it: _______________________

    The one-line test that would have caught it: _______________________________

88. Technology. How to confirm your graph of $5x - 2y < 10$, and the error a disagreement most likely points at:

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PAGE 29 — Explaining and interpreting

Say How, and Say What It Means

91. Explain the method. How to graph $2x - y > 4$ — reversal, line style, test point, confirming substitution.

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    ---

    ---

92. Explain. Why does one successful substitution not prove your shading is right? Which check does?

    ---

93. Apply it. Candidate A of the figure, interpreted with units: _______________________________

94. Apply it. Verify $(3,8)$: hours _______________________ earnings _______________________

    Interpreted, with dollars: _______________________________________

PAGE 30 — Corners, and exit ticket 9.5

What a Corner Means

89. The corner $(3,1)$ of $y \ge x - 2$, $y < -x + 4$.

    In $y = x - 2$: _______________________ In $y = -x + 4$: _______________________

    A solution of the system? ______ Why? _______________________

90. Apply it. The corner $(12,0)$, interpreted with units: _______________________________

    Why corners are worth asking about: _______________________________

91. Apply it. $(8,5)$ in $5x + 8y \le 80$: _______________________ It means _______________________

Exit Ticket · Lesson 9.5

93. $(2,3)$ in $y \ge x - 1$, $y < 4$: ______ Substitutions: _______________________

94. $(-1,5)$ in that system: ______ Inequality it fails: ____________

95. Explain. The three ways to verify, and what each catches:

    ---

96. Apply it. One solution of $12x + 9y \ge 180$: ____________ Interpreted, with units: _______________________

PAGE 31 — Chapter 9 review · graphing one inequality

Chapter 9 Review

Part A · Graphing one linear inequality

97. $y \le -2x + 4$ Boundary: ____________ Style: ______ Test point: ______ Shade: ____________

98. $5x - 2y < 10$ Solved for $y$: ____________ Reversing step: _______________________

    Origin test in the original: _______________________ Shade: ____________

99. $x \le 3$ Boundary: ____________ Region: ____________________

100. Solid boundary through $(0,4)$ and $(2,0)$, region containing the origin.

     Inequality: ____________ Check: _______________________

FIGURE: fig3-dashed-versus-solid-boundary.png (half width)

101. Using $(2,3)$, what the two line styles mean: _______________________________________

102. Find the error. A student rewrites $2x - y > 4$ as $y > 2x - 4$ and shades above.

     Error: _______________________ What $(0,0)$ in the original shows: _______________________

PAGE 32 — Chapter 9 review · writing inequalities

Chapter 9 Review (continued)

Part B · Creating an inequality from a situation

103. Apply it. A truck carries at most $1{,}200$ lb; crates $50$ lb, bags $30$ lb.

     $x$ is ____________________ $y$ is ____________________ Inequality: _______________________

104. Apply it. $20$ crates and $8$ bags? ______ Arithmetic: ____________ It means _______________________

FIGURE: fig7-budget-inequality-in-context.png (half width)

105. Apply it. Interpret each, with units.

106. Explain. Why the first quadrant, and what that does to the region:

     ---

107. Apply it. Tickets $\$25$, sponsorships $\$40$, more than $\$500$: _______________________

     $x$ is ____________________ $y$ is ____________________ Boundary: ____________

108. Explain. "at least $\$500$" versus "more than $\$500$" — symbol, boundary, and one amount they disagree about:

     ---

PAGE 33 — Chapter 9 review · systems

Chapter 9 Review (continued)

Part C · Systems of two linear inequalities

109. Graph $y \ge -x + 2$ and $y < 3x - 2$. Corner: ____________

110. Is $(4,2)$ a solution? ______ Both substitutions: _______________________

FIGURE: fig9-overlap-region-and-corner.png (half width)

111. The system: ____________________ and ____________________ Corner: ____________

     Is the corner a solution? ______ Why? _______________________

112. Apply it. The work-schedule system.

     System: ____________________ and ____________________

     $x$ is ____________________ $y$ is ____________________

     Corners: ____________ ____________ ____________

     $(0,8)$ means _______________________________________

113. A system with no solutions: ____________________ Why the graph shows it: _______________________

114. Explain. Why a system of inequalities gives a region with corners while a system of equations gives a point:

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PAGE 34 — Chapter 9 review · verifying

Chapter 9 Review (continued)

Part D · Verifying and interpreting

115. Apply it. $(6,4)$ in $x + y \le 12$, $8x + 12y \ge 96$.

     Hours: _______________________ Earnings: _______________________ Solution? ______

     Interpreted, with units: _______________________________________

116. $(2,9)$ in $y < 5x$ and $y \ge x + 2$: ______ Substitutions: _______________________

     How a graph would confirm: _______________________________

117. Apply it. $5x + 8y \le 80$ with exactly $5$ game passes.

     Inequality that results: ____________ Whole-number ride bands possible: _______________________

     Interpreted: _______________________________________

PAGE 35 — Graph paper

Graph Paper

FIGURE: fig12-blank-grids.png (full page)

Use these grids for any inequality or system a teacher assigns alongside this chapter.

Three reminders for every sketch:

• Draw the boundary first, and make the dashed or solid decision before you shade anything.
• Test a point that is not on the boundary — the origin whenever the boundary misses it — and write the substitution beside the graph.
• For a system, shade each inequality lightly in a different direction or color, then outline the overlap, and mark the corner where the two boundaries meet.

Canva production notes

• Page size: 8.5 × 11 in, 0.75 in margins
• Type: headings 24–28 pt, body 12–14 pt, answer blanks 14 pt with 1.5 line spacing
• Item order: a few items are placed by figure rather than by number, so that every item on a page can be answered from the figure on that page. Page 3 runs 3–6 then 14; page 4 runs 9 then 17; page 9 pairs 23 with 28; page 10 pairs 24 with 29; page 27 runs 83, 84, then 80; page 28 runs 85, 81, 88, then 117. The numbers still match the textbook exactly, and the answer key is in numerical order.
• Figure widths: fig1-interval-versus-region.png, fig3-dashed-versus-solid-boundary.png, fig4-shade-above-or-below.png, and fig6-horizontal-and-vertical-boundaries.png are two-panel images — set them full width and do not place text beside them, or the panels shrink below readable. fig7, fig10, and fig11 carry axis labels with units and must not be reduced below half a page, or the unit labels stop being legible. fig12-blank-grids.png takes a full page on its own and repeats on pages 8, 11, 22, and 35.
• The shading has to survive printing. Every region in this chapter is a light tint. Do not print these pages in a low-toner draft mode: a figure whose shading is invisible turns every graphing item into a guess. If shading reproduces poorly, have students hatch the region with pencil lines instead of coloring it in.
• Two colors for systems. On pages 20–25, students shade two half planes on one grid. Provide two colored pencils, or ask for hatching in two directions — the overlap must be visibly different from either single region, since that overlap is the answer.
• Blank work space: items 4, 10, 12, 17, 18, 26, 32, 38, 40, 48, 49, 51, 55, 59, 61, 65, 66, 70, 75, 79, 81, 85, 87, 88, 91, 95, 101, 102, 106, 108, 113, 114, and 117 are written explanations — give each at least four ruled lines. Item 91 is a full paragraph and needs at least eight.
• Graphing items need a grid, not a blank: items 21, 22, 25, 27, 28, 29, 35, 36, 37, 63, 73, 97, 98, 99, and 109 all end in a drawn region. Place a grid beside each, or route the student to the graph-paper page and number the grid used.
• Fractions: items 16, 22, 25c, 63d, and 98 have fractional slopes. Set them as stacked fractions, not slashes.
• Negative signs: use the typographic minus (−), matching the figures, not a hyphen.
• Currency: dollar amounts appear on pages 14–19, 24, 25, 29, 32, and 34. Set them as plain text ($5, $8, $9, $12, $25, $40, $50, $60, $80, $96, $140, $180, $200, $500), not inside a math box.
• Ordered pairs: every blank for a solution, a corner, or an intercept must be wide enough for an ordered pair with a comma and parentheses. A short blank invites a single number, which is this chapter's most common error.