Appendix A — Answer Key, Chapter 12: Inequalities in One Variable

SOL 6.PFA.4 · Covers textbook Chapter 12 and the companion workbook. Item numbers match the textbook; where the workbook repeats the same problems, this key serves both, and workbook-only items are listed at the end. Reasoning answers show an acceptable response, not the only wording. Variable definitions may use any letter as long as the quantity and its units are stated.

Lesson 12.1 — What an Inequality Says

Guided practice

1. a) Let $a$ be the age in years; $a \ge 13$. b) Let $w$ be the weight in pounds; $w < 20$.
2. No. Substituting gives $9 > 9$, which is false — a number is not greater than itself.
3. Yes. Substituting gives $9 \ge 9$, which is true because of the "or equal to" part.
4. Every number that is $4$ or less, including $4$ itself.
5. Let $p$ be the number of people; $p \le 60$.

Independent practice

6. a) Let $t$ be the temperature in °F; $t > 32$. b) Let $c$ be the cost in dollars; $c \le 75$. c) Let $n$ be the number of tickets sold; $n \ge 100$.
7. a) $x$ is less than six. b) $y$ is greater than or equal to negative two. c) $t$ is less than or equal to ten.
8. Solutions: any three numbers greater than $5$, such as $6$, $10$, and $100$. Not a solution: any number $5$ or less, such as $5$ or $2$.
9. In $w < 12$ the value $12$ is not allowed; in $w \le 12$ it is. Every other number behaves the same way in both, so $12$ is the only difference between the two solution sets.
10. Let $t$ be the internal temperature in °F; $t \ge 165$. Yes, 165°F is done enough, because $165 \ge 165$ is true.
11. Let $w$ be the total weight in pounds; $w \le 2{,}500$. Exactly 2,500 pounds is permitted, since $2{,}500 \le 2{,}500$ is true. For 2,500 pounds to be too heavy, the sign would have to read something like "load must be under 2,500 pounds," which gives the strict inequality $w < 2{,}500$.
12. False. The solution set of $x \le 99$ stops at $99$, while $x < 100$ accepts every number below $100$. For example, $99.5$ satisfies $99.5 < 100$ but not $99.5 \le 99$.

Exit ticket 12.1

1. $w \le 250$
2. Yes. Substituting gives $-3 \ge -3$, which is true because the two sides are equal.
3. Every number greater than $7$, not including $7$ itself.
4. An inequality describes a whole range of acceptable values rather than one exact value, so every number on the accepted side of the boundary makes it true.

Lesson 12.2 — The Four Inequality Symbols

Guided practice

1. a) "$x$ is less than three"; the boundary $3$ is not in the solution set. b) "$x$ is greater than or equal to three"; the boundary $3$ is in the solution set.
2. $s \le 55$
3. $v \ge 4$
4. $b > 0$
5. Strict: $<$ and $>$. Inclusive: $\le$ and $\ge$.

Independent practice

6. a) $g < 8$ b) $p \ge 15$ c) $n \le 3$ d) $d < -20$, where $d$ is the elevation in feet relative to sea level (deeper means a lesser elevation)
7. a) $m$ is less than or equal to negative six. b) $k$ is greater than zero. c) $y$ is greater than or equal to twenty-one.
8. a) boundary $15$; not a solution b) boundary $15$; a solution c) boundary $-9$; a solution
9. For example $x < 30$ and $x \le 30$. The two solution sets are identical except for the value $30$, which belongs only to the inclusive one.
10. Let $a$ be the age in years; $a < 5$. The symbol is strict because a child who has already turned 5 is no longer "under 5" and does not enter free.
11. Let $b$ be the number of boxes packed in a shift; $b \ge 25$. Let $p$ be the weight lifted at one time in pounds; $p \le 40$. Both include their boundary values: "no fewer than 25" allows exactly 25, and "no more than 40" allows exactly 40. To exclude 40, the rule would have to read "no one may lift 40 pounds or more" or "everything lifted must weigh under 40 pounds," giving $p < 40$.
12. $x \le 8$ is defined to mean that at least one of $x < 8$ and $x = 8$ holds, so the two descriptions accept exactly the same numbers. $x < 8$ alone leaves out the single value $x = 8$, so its solution set is smaller.

Exit ticket 12.2

1. $p \le 40$
2. "$x$ is greater than or equal to negative seven."
3. $\le$ and $\ge$
4. "No more than 12" says 12 is the largest acceptable value, so 12 itself is acceptable. That requires the inclusive symbol $\le$. Writing $x < 12$ would wrongly rule 12 out.

Lesson 12.3 — Graphing Inequalities on a Number Line

Guided practice

1. Open circle at $5$, shaded left with an arrowhead.
2. Closed circle at $0$, shaded right with an arrowhead.
3. $x \le 4$
4. $x > -6$
5. Because the solution set never ends in that direction. The arrowhead says the shading continues past the edge of the drawing.

Independent practice

6. a) open circle at $-4$, shaded right b) closed circle at $7$, shaded left c) closed circle at $-2$, shaded right d) open circle at $0$, shaded left
7. a) $x < 10$ b) $x \ge -8$ c) $x \le 3$
8. The graphs are identical except at the boundary: $x \le 6$ has a closed circle at $6$ because $6$ is a solution, and $x < 6$ has an open circle because it is not.
9. Let $g$ be the number of gallons; $g \le 15$. Closed circle at $15$, shaded left with an arrowhead.
10. For example: during a test dive, a submarine may not go below 5 feet under the surface, so its elevation $x$ in feet satisfies $x \ge -5$. The boundary value $-5$ is the lowest allowed elevation, and it is allowed.
11. Let $a$ be the age in years; $a \ge 10$. Closed circle at $10$, shaded right with an arrowhead. The closed circle means a swimmer who is exactly 10 years old may use the deep end — the boundary age counts.
12. A closed circle claims that $3$ itself is a solution, but $x > 3$ is strict. The value $3$ exposes the error, since $3 > 3$ is false. The circle should be open.

Exit ticket 12.3

1. Closed circle at $-1$, shaded left with an arrowhead.
2. $x > 7$
3. $<$ and $>$
4. A filled circle means the boundary value is included in the solution set; a hollow circle means it is excluded. The circle is the only part of the graph that answers that question.

Mid-chapter check (Lessons 12.1–12.3)

1. $p \ge 9$
2. $w < 4$
3. Boundary value $-12$; it is a solution, since $-12 \ge -12$ is true.
4. Open circle at $0$, shaded right with an arrowhead.
5. $x \le -4$
6. No. Substituting gives $6 < 6$, which is false.
7. "$t$ is less than or equal to twenty."
8. For example: a club must collect at least $30 to cover supplies, so the amount collected $c$ in dollars satisfies $c \ge 30$.

Lesson 12.4 — Writing an Inequality Two Equivalent Ways

Guided practice

1. $6 < x$
2. $-2 \ge x$
3. $m > 7$
4. $k \le -4$
5. Because the symbol reports which quantity is greater. Swapping the sides moves the greater quantity to the other side, so the symbol has to turn around for the statement to stay true.

Independent practice

6. a) $11 > x$ b) $-6 \le x$ c) $y < 0$ d) $n \ge -15$
7. $x < 4$ and $4 > x$
8. $x \ge -7$ and $-7 \le x$
9. $c \le 25$; closed circle at $25$, shaded left with an arrowhead.
10. Yes, they are equivalent. The value $0$ satisfies both $0 > -8$ and $-8 < 0$. The value $-20$ satisfies neither, since $-20 > -8$ and $-8 < -20$ are both false. Every number behaves the same way in both statements.
11. $p \le 220$. In plain English: no more than 220 people may be inside. Exactly 220 people is allowed, since $220 \le 220$ is true.
12. $x \le 9$ and $9 \ge x$ agree on every value: at $x = 9$ both say $9$ compared with $9$ and both are true, and at $x = 2$ both are true as well. But $9 \le x$ is a different statement: at $x = 9$ it is true, while at $x = 2$ it becomes $9 \le 2$, which is false even though $2 \le 9$ is true. Swapping sides without reversing the symbol changes the solution set.

Exit ticket 12.4

1. $5 \le x$
2. $w > -2$
3. $x < 3$ and $3 > x$
4. The symbol's point faces the lesser quantity and its open end faces the greater one. Swapping the sides exchanges those positions, so the symbol must reverse to keep facing the same quantities.

Lesson 12.5 — Testing Values and Describing Solution Sets

Guided practice

1. Yes. $5 \ge 5$ is true.
2. No. $5 > 5$ is false.
3. Only $0$. Substituting gives $0 < 2$ true, $2 < 2$ false, $6 < 2$ false.
4. Any number greater than $10$, such as $12$.
5. Yes. Graphing $x \le -4$ gives a closed circle at $-4$ shaded left, and $-7$ lies to the left of $-4$, inside the shaded region. By substitution, $-7 \le -4$ is true.

Independent practice

6. a) $-5 \ge -3$ is false; not a solution. b) $-3 \ge -3$ is true; a solution. c) $0 \ge -3$ is true; a solution. d) $4 \ge -3$ is true; a solution.
7. Only $-8$. Substituting: $-8 < -2$ true; $-2 < -2$ false; $1 < -2$ false; $9 < -2$ false.
8. Any three numbers that are $0$ or less, such as $-6$, $-1$, and $0$.
9. $-5$. The boundary $-6$ is excluded because the symbol is strict, and $-5$ is the next integer to the right; $-5 > -6$ is true.
10. $12$. The symbol is inclusive, so the boundary itself qualifies: $12 \le 12$ is true, and every integer above $12$ fails.
11. $L \le 36$. The 30-inch box ships, since $30 \le 36$ is true. The 36-inch box ships, since $36 \le 36$ is true. The 40-inch box does not, since $40 \le 36$ is false.
12. On a number line $-12$ lies far to the left of $-5$, so it is less than $-5$, not greater; $-12 > -5$ is false. He compared the digits 12 and 5 instead of the positions. A value that is a solution: any number greater than $-5$, such as $0$.

Exit ticket 12.5

1. Yes. Substituting gives $-1 \ge -1$, which is true.
2. Only $-6$. Substituting: $-6 < 0$ true; $0 < 0$ false; $4 < 0$ false.
3. Any number that is $-8$ or less, such as $-10$.
4. Graph the inequality, then locate the number. If it lands outside the shaded region — or sits on an open circle — it is not a solution.

Chapter 12 Review

Part A — Two equivalent ways from a graph (6.PFA.4a)

1. $x < -5$ and $-5 > x$
2. $x \ge 6$ and $6 \le x$
3. $x \le -2$ and $-2 \ge x$
4. $y > 18$; open circle at $18$, shaded right with an arrowhead.

Part B — Writing an inequality from a constraint or graph (6.PFA.4b)

5. a) Let $p$ be the number of passengers; $p \le 12$. b) Let $t$ be the water temperature in °F; $t > 68$. c) Let $n$ be the number of cans collected; $n \ge 40$.
6. Let $h$ be the height in inches; $h < 48$.
7. $x > 0$
8. $c \le 85$

Part C — Creating a graph or a context (6.PFA.4c)

9. a) open circle at $-1$, shaded left b) closed circle at $4$, shaded right c) closed circle at $0$, shaded left
10. For example: a carry-on bag may weigh at most 25 pounds, so its weight $w$ in pounds satisfies $w \le 25$. The boundary value 25 is the heaviest weight allowed, and it is allowed.
11. For example: a petition needs at least 100 signatures to be filed, so the number of signatures $n$ satisfies $n \ge 100$.
12. Rewritten with the variable first: $x \ge -3$. Closed circle at $-3$, shaded right with an arrowhead.

Part D — Justifying with substitution or a graph (6.PFA.4d)

13. Yes. Substituting gives $4 \le 4$, which is true.
14. No. The graph of $x > -9$ has an open circle at $-9$ shaded right, and $-11$ lies to the left of $-9$, outside the shaded region. By substitution, $-11 > -9$ is false.
15. $-3$, $0$, and $5$. Substituting: $-7 \ge -3$ false; $-3 \ge -3$ true; $0 \ge -3$ true; $5 \ge -3$ true.

Part E — Identifying values in a solution set (6.PFA.4e)

16. Any two numbers greater than $9$, such as $10$ and $25$.
17. $-14$, because the inclusive symbol makes the boundary itself a solution.
18. $6$, because $7$ is excluded by the strict symbol and $6$ is the next integer down.

Part F — Mixed application and reasoning

19. Let $h$ be the height in inches; $h \ge 48$, equivalently $48 \le h$. Graph: closed circle at $48$, shaded right with an arrowhead. A swimmer exactly 48 inches tall may use the board, since $48 \ge 48$ is true.
20. The two statements are the same fact read from opposite ends. Swapping the sides of $x \ge -6$ requires reversing $\ge$ to $\le$, which produces $-6 \le x$. Both accept exactly the numbers $-6$ and greater, so they have the same solution set and both match the graph.
21. The two solution sets are identical for every number below $20$. They differ only at $20$ itself, which belongs to $x \le 20$ but not to $x < 20$.
22. Let $w$ be the total weight in pounds; $w \le 1{,}200$. A weight in the solution set: $900$, since $900 \le 1{,}200$ is true. A weight not in the solution set: $1{,}500$, since $1{,}500 \le 1{,}200$ is false.

Workbook-only items

Page 2, match the phrase. is more than → $>$; is at most → $\le$; is fewer than → $<$; is at least → $\ge$; cannot exceed → $\le$; is no less than → $\ge$

Page 2, constraints table.

Page 3, boundary table. $x > 9$: boundary $9$, not a solution. $x \ge 9$: boundary $9$, a solution. $t \le 10$: boundary $10$, a solution. $t < 10$: boundary $10$, not a solution. $y \ge -2$: boundary $-2$, a solution.

Page 3, in words. $x < 6$: $x$ is less than six. $y \ge -2$: $y$ is greater than or equal to negative two. $t \le 10$: $t$ is less than or equal to ten.

Page 3, solutions of $x > 5$. Any three numbers greater than $5$, such as $6$, $10$, $100$. Not a solution: any number $5$ or less, such as $5$.

Page 3, explain. The value $12$ is a solution of $w \le 12$ but not of $w < 12$. That single value is the only difference between the two solution sets.

Page 5, fill in the symbol. a) $<$ b) $\ge$ c) $\le$ d) $\le$ e) $\ge$ f) $>$ g) $<$ h) $\ge$

Page 5, watch out. "No more than 70" is written $w \le 70$, not $w < 70$, because 70 is the largest acceptable value and is therefore acceptable.

Page 6, translation repair.

Page 6, two rules. Boxes: $b \ge 25$. Lifting: $p \le 40$. Both include their boundary values, because "no fewer than 25" allows exactly 25 boxes and "no more than 40" allows exactly 40 pounds. To exclude 40 pounds, the rule could read "everything lifted must weigh under 40 pounds," giving $p < 40$.

Page 8, graphing table.

Page 9, graph these. 1. $x \ge 1$: closed circle at $1$, shaded right with an arrowhead. 2. $x < -2$: open circle at $-2$, shaded left with an arrowhead.

Page 9, inequality for the figure. $x \le -5$

Page 9, read the graphs. 3. $x < 10$ 4. $x \ge -8$ 5. $x \le 3$

Page 9, create a context. For example: during a test dive, a submarine may not go below 5 feet under the surface, so its elevation $x$ in feet satisfies $x \ge -5$. The boundary value $-5$ is the lowest allowed elevation, and it is allowed.

Page 11, mid-chapter check. Same answers as the textbook mid-chapter check above.

Page 12, complete the pairs.

Page 12, write each the other way. $x < 11$ becomes $11 > x$. $x \ge -6$ becomes $-6 \le x$. $0 > y$ becomes $y < 0$. $25 \ge c$ becomes $c \le 25$.

Page 13, graph to both forms.

Page 13, careful. Rewrite $-5 > x$ as $x < -5$, then shade to the left. Shading right would follow the shape of the symbol instead of its meaning: the open end faces the greater quantity, which here is $-5$, not $x$.

Page 13, apply it. Variable first: $p \le 220$. In plain English: no more than 220 people may be inside. Exactly 220 people is allowed.

Page 15, testing table for $x \ge -3$.

Page 15, circle the solutions of $x < -2$. Only $-8$.

Page 15, careful with negatives. No, $-12$ is not a solution of $x > -5$. On a number line $-12$ lies to the left of $-5$, so it is less than $-5$, and the graph's shading (right of an open circle at $-5$) does not cover it.

Page 16, name a value. $x > 9$: any number greater than $9$, such as $12$. $x \le -8$: any number $-8$ or less, such as $-10$. $x \ge 0$: any number $0$ or greater, such as $5$. $x < -1$: any number less than $-1$, such as $-4$.

Page 16, least and greatest. Least integer for $x > -6$: $-5$, because $-6$ is excluded by the strict symbol. Greatest integer for $x \le 12$: $12$, because the inclusive symbol keeps the boundary. Least integer for $x \ge -14$: $-14$. Greatest integer for $x < 7$: $6$.

Page 16, shipping table. Inequality: $L \le 36$.

Pages 18–20, Chapter 12 review. Same answers as the textbook Chapter 12 Review above.