Grade 6 Workbook — Chapter 12: Inequalities in One Variable

SOL 6.PFA.4 · Companion to Textbook Chapter 12

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/.

PAGE 1 — Chapter opener

Chapter 12 · Inequalities in One Variable

Standard 6.PFA.4

In this chapter you will:

• Write an inequality for a constraint in context or from a graph
• Use the symbols $<$, $>$, $\le$, and $\ge$
• Graph an inequality on a number line with an open or closed circle
• Write the same inequality two equivalent ways
• Test values and name numbers in the solution set

Words to know: inequality · solution · solution set · constraint · boundary value · open circle · closed circle · equivalent inequalities

Remember: an equation has one answer. An inequality usually has many.

PAGE 2 — What an inequality says

12.1 What an Inequality Says

An inequality compares two quantities that are not necessarily equal. A solution is any value that makes it true. All the solutions together are the solution set.

Match each phrase to its symbol. Write $<$, $>$, $\le$, or $\ge$.

Write an inequality for each constraint. Define the variable first.

PAGE 3 — Does the boundary count?

The Boundary Value

The one question that decides the symbol: is the boundary number itself allowed?

Answer yes or no for each.

Write each inequality in words.

$x < 6$: _______________________________________________

$y \ge -2$: _______________________________________________

$t \le 10$: _______________________________________________

Name three solutions of $x > 5$ and one number that is NOT a solution.

Solutions: ______, ______, ______ Not a solution: ______

Explain. How are the constraints $w < 12$ and $w \le 12$ different? Use the value $12$ in your answer.

PAGE 4 — Exit ticket 12.1

Exit Ticket · Lesson 12.1

Name: ________________________ Date: ____________

1. Write an inequality: a rider must weigh at most 250 pounds. Use $w$. ______

2. Is $-3$ a solution of $x \ge -3$? ______ Explain.

3. Describe the solution set of $n > 7$ in words.

4. Why does an inequality usually have more than one solution?

PAGE 5 — The four symbols

12.2 The Four Inequality Symbols

FIGURE: fig1-four-symbols-reference.png (full width)

[Reference card in four cells: each cell shows one symbol large, its name in words, one example inequality using x, and whether the boundary is included. The extra bar under the inclusive symbols is highlighted in a second color.]

Fill in $<$, $>$, $\le$, or $\ge$.

Watch out. "No more than 70" is written ______________, not ______________.

Why?

PAGE 6 — Fix the translation

Translation Repair

Three of these translations are WRONG. Mark each right or wrong, and correct the wrong ones.

Two rules, two symbols. A food bank asks each volunteer to pack no fewer than 25 boxes per shift, and no one may lift more than 40 pounds at a time.

Boxes: ______________ Lifting: ______________

Both rules include their boundary values. Explain why.

Rewrite the lifting rule in words so that 40 pounds is NOT allowed.

PAGE 7 — Exit ticket 12.2

Exit Ticket · Lesson 12.2

Name: ________________________ Date: ____________

1. Fill in the symbol: the room holds at most 40 people. $p \ \square \ 40$

2. Read $x \ge -7$ aloud in words.

3. Which symbols include the boundary value? ______________

4. Why is "no more than 12" written $x \le 12$ instead of $x < 12$?

PAGE 8 — Open circle or closed circle

12.3 Graphing on a Number Line

FIGURE: fig2-open-vs-closed-circle.png (full width)

[Two identical number lines from -2 to 8. Top: open circle at 3 shaded right, labeled x > 3, captioned "3 is not a solution." Bottom: closed circle at 3 shaded right, labeled x >= 3, captioned "3 is a solution."]

Open circle = boundary NOT included ($<$, $>$) Closed circle = boundary IS included ($\le$, $\ge$) Arrowhead = the solutions keep going forever

FIGURE: fig3-graphing-four-symbols.png (half width, right side)

[Four short number lines from -5 to 5 in a 2 by 2 grid, headed x < 1, x <= 1, x > -2, x >= -2, each with the correct circle type and shading direction.]

Complete the table.

PAGE 9 — Graph it, then read it back

Graphing Both Directions

Graph each inequality on the number lines below.

1. $x \ge 1$

BLANK NUMBER LINE: -6 to 6, integer ticks labeled

2. $x < -2$

BLANK NUMBER LINE: -6 to 6, integer ticks labeled

Now go the other way. Write the inequality for each described graph.

FIGURE: fig4-read-graph-to-inequality.png (full width)

[Number line from -8 to 2 with a closed circle at -5 shaded left to an arrowhead. Callout labels point to the boundary value, the filled circle ("closed: included"), and the shaded ray ("direction: lesser numbers"). No inequality printed.]

Inequality for the figure: ______________

3. Open circle at $10$, shaded left: ______________

4. Closed circle at $-8$, shaded right: ______________

5. Closed circle at $3$, shaded left: ______________

Create a context. Write a real situation that matches $x \ge -5$ exactly.

What does the boundary value mean in your situation?

PAGE 10 — Exit ticket 12.3

Exit Ticket · Lesson 12.3

Name: ________________________ Date: ____________

1. Describe the graph of $x \le -1$.

2. A graph has an open circle at $7$ and shades right. Write the inequality. ______________

3. Which symbols get an open circle? ______________

4. How does the circle tell a reader whether the boundary value is a solution?

PAGE 11 — Mid-chapter check

Mid-Chapter Check · Lessons 12.1–12.3

Name: ________________________ Date: ____________

1. The team needs at least 9 players. ______________

2. The package weighs less than 4 pounds. ______________

3. Boundary value of $x \ge -12$: ______ Is it a solution? ______

4. Describe the graph of $x > 0$.

5. Closed circle at $-4$, shaded left. Inequality: ______________

6. Is $6$ a solution of $x < 6$? ______ Why?

7. Write $t \le 20$ in words. _______________________________________________

8. Create a situation for $c \ge 30$. _______________________________________________

PAGE 12 — Two equivalent ways

12.4 Two Equivalent Ways

FIGURE: fig5-two-equivalent-ways.png (full width)

[Number line from -9 to 1 with an open circle at -5 shaded left. Below it, two boxes joined by a double-headed arrow labeled "same solution set": left box "x < -5" captioned "start with the variable," right box "-5 > x" captioned "start with the number." One line under both: "swap the sides, reverse the symbol."]

Swap the sides, reverse the symbol. The bar on $\le$ and $\ge$ stays put.

Complete the pairs.

Write each the other way.

$x < 11$ becomes ______________

$x \ge -6$ becomes ______________

$0 > y$ becomes ______________

$25 \ge c$ becomes ______________

PAGE 13 — Graphs to two forms

From Graph to Both Forms

Write each inequality TWO equivalent ways.

Careful. To graph $-5 > x$, first rewrite it as ______________, then shade to the ______________.

Why is it wrong to shade right just because the symbol's open end points right?

Apply it. A theater's fire code is posted as $220 \ge p$, where $p$ is the number of people inside.

Variable first: ______________ In plain English: _______________________________________________

Is exactly 220 people allowed? ______

PAGE 14 — Exit ticket 12.4

Exit Ticket · Lesson 12.4

Name: ________________________ Date: ____________

1. Rewrite $x \ge 5$ with the number first. ______________

2. Rewrite $-2 < w$ with the variable first. ______________

3. Open circle at $3$, shaded left. Write it two ways: ______________ and ______________

4. Why must the symbol reverse when the sides are swapped?

PAGE 15 — Testing values

12.5 Testing Values

FIGURE: fig6-testing-values-on-a-graph.png (full width)

[Number line from -8 to 4 with a closed circle at -5 shaded right to an arrowhead. Test values marked above the line: -7 with a cross, -5 with a check, -2 with a check, 3 with a check. Caption: "x >= -5: values in the shaded region are solutions."]

Two ways to justify: substitute, or check the graph.

Complete the table for $x \ge -3$.

Circle the values that are solutions of $x < -2$.

$$-8 \qquad -2 \qquad 1 \qquad 9$$

Careful with negatives. Is $-12$ a solution of $x > -5$? ______ Explain using the number line.

PAGE 16 — Describing solution sets

Naming Values in the Solution Set

Name a value in each solution set.

Least and greatest.

Least integer in the solution set of $x > -6$: ______ Why? _______________________

Greatest integer in the solution set of $x \le 12$: ______ Why? _______________________

Least integer in the solution set of $x \ge -14$: ______

Greatest integer in the solution set of $x < 7$: ______

Apply it. A shipping rule says a box's longest side must be at most 36 inches. Inequality: ______________

PAGE 17 — Exit ticket 12.5

Exit Ticket · Lesson 12.5

Name: ________________________ Date: ____________

1. Is $-1$ a solution of $x \ge -1$? ______ Substitution: ______________

2. Circle the solutions of $x < 0$: $-6$ · $0$ · $4$

3. Name one value in the solution set of $x \le -8$. ______

4. How can a number line graph show that a number is NOT a solution?

PAGE 18 — Chapter 12 review, part 1

Chapter 12 Review

Part A · Two equivalent ways from a graph

1. Open circle at $-5$, shaded left: ______________ and ______________

2. Closed circle at $6$, shaded right: ______________ and ______________

3. Closed circle at $-2$, shaded left: ______________ and ______________

4. Rewrite $18 < y$ with the variable first: ______________ Describe its graph: _______________________

Part B · Writing an inequality from a constraint or graph

5. a) The elevator holds at most 12 passengers: ______________

   b) The water temperature stayed above 68°F: ______________

   c) At least 40 cans were collected: ______________

6. "Under 48 inches must ride with an adult." Inequality: ______________

7. Open circle at $0$, shaded right, variable first: ______________

8. Spending of no more than $85, using $c$: ______________

PAGE 19 — Chapter 12 review, part 2

Chapter 12 Review (continued)

Part C · Creating a graph or a context

9. Describe the graph: a) $x < -1$ _______________________ b) $x \ge 4$ _______________________

   c) $x \le 0$ _______________________

10. Situation for $w \le 25$: _______________________________________________

    What does the boundary value mean? _______________________________________________

11. Situation for $n \ge 100$: _______________________________________________

12. Rewrite $-3 \le x$ with the variable first: ______________ Describe its graph: _______________________

Part D · Justifying with substitution or a graph

13. Is $4$ a solution of $x \le 4$? ______ Substitution: ______________

14. Is $-11$ a solution of $x > -9$? ______ Justify with the number line.

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15. Circle the solutions of $x \ge -3$: $-7$ · $-3$ · $0$ · $5$

PAGE 20 — Chapter 12 review, part 3

Chapter 12 Review (continued)

Part E · Identifying values in a solution set

16. Two values in the solution set of $x > 9$: ______ and ______

17. Least integer in the solution set of $x \ge -14$: ______

18. Greatest integer in the solution set of $x < 7$: ______

Part F · Application and reasoning

19. Swimmers must be at least 48 inches tall to use the diving board.

    Inequality: ______________ Other equivalent way: ______________

    Graph: _______________________ May a 48-inch swimmer use the board? ______

20. One student writes $x \ge -6$ for a closed circle at $-6$ shading right; another writes $-6 \le x$. Why are both correct?

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21. How do the solution sets of $x < 20$ and $x \le 20$ differ? Name the one value that separates them. ______

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22. A van may carry no more than 1,200 pounds. Inequality: ______________

    One weight in the solution set: ______ Substitution: ______________

    One weight not in the solution set: ______ Substitution: ______________

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
• Blank number lines: page 9 needs two labeled number lines from $-6$ to $6$ with integer ticks and room above the line for circles and shading; leave at least 0.5 in of vertical clearance
• Figure widths: all number-line figures full width except fig3-graphing-four-symbols.png, which is a 2 by 2 grid and sits at half width
• Circles: open circles must be clearly hollow at print size — use a 2 pt stroke with white fill, never gray
• Answer blanks: keep every blank on the same line as its prompt so the Canva text box does not reflow