Middle School Math Quiz: Solve And Graph Linear Inequalities
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Solve And Graph Linear InequalitiesQuestion 1 of 20

A taxi charges $3.50 for the first mile plus $2.25 for each additional mile. Sarah has no more than $20 to spend. What is the maximum distance she can travel?

66 miles
77 miles
88 miles
99 miles
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Middle School Math Quiz

Middle School Math Quiz: Solve And Graph Linear Inequalities

Practice Solve And Graph Linear Inequalities in Middle School Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Solve And Graph Linear Inequalities, giving you a quick way to practice the rules, question types, and explanations that matter most for Middle School Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

A taxi charges $3.50 for the first mile plus $2.25 for each additional mile. Sarah has no more than $20 to spend. What is the maximum distance she can travel?

  1. 66 miles
  2. 77 miles
  3. 88 miles (correct answer)
  4. 99 miles
Explanation: For mm miles where m>1m > 1, the cost is 3.50+2.25(m−1)3.50 + 2.25(m-1). The inequality is 3.50+2.25(m−1)≤203.50 + 2.25(m-1) \leq 20. Expanding: 3.50+2.25m−2.25≤203.50 + 2.25m - 2.25 \leq 20, so 1.25+2.25m≤201.25 + 2.25m \leq 20, giving 2.25m≤18.752.25m \leq 18.75, thus m≤8.33m \leq 8.33. Since we need whole miles, the maximum is 88 miles. Checking: 88 miles costs 3.50 + 2.25(7) = \19.25 \leq $20.For. For 9miles:miles:3.50 + 2.25(8) = $21.50 > $20$, which exceeds the budget.

Question 2

A phone plan charges $40 per month plus $0.15 per text message over $200200 $ messages. If Chen's monthly bill cannot exceed $55, what is the maximum number of text messages he can send?

  1. Chen can send at most 300300 text messages per month (correct answer)
  2. Chen can send at most 250250 text messages per month
  3. Chen can send at most 200200 text messages per month
  4. Chen can send at most 350350 text messages per month
Explanation: If Chen sends tt messages where t>200t > 200, his bill is 40+0.15(t−200)40 + 0.15(t - 200). The inequality is 40+0.15(t−200)≤5540 + 0.15(t - 200) \leq 55, giving 0.15(t−200)≤150.15(t - 200) \leq 15, so t−200≤100t - 200 \leq 100, thus t≤300t \leq 300. Choice B (250250) is unnecessarily restrictive. Choice C (200200) ignores that he can go over the base amount. Choice D (350350) would cost 40+0.15(150)=$62.5040 + 0.15(150) = \$62.50, exceeding his budget.

Question 3

A student's final grade is calculated as 0.7T+0.3F0.7T + 0.3F where TT is the test average and FF is the final exam score. If a student has a test average of 7878 and needs a final grade of at least 8282, what minimum score is needed on the final exam?

  1. The student needs at least 8888 points on the final exam
  2. The student needs at least 9090 points on the final exam
  3. The student needs at least 9292 points on the final exam (correct answer)
  4. The student needs at least 9494 points on the final exam
Explanation: Setting up the inequality: 0.7(78)+0.3F≥820.7(78) + 0.3F \geq 82. This gives 54.6+0.3F≥8254.6 + 0.3F \geq 82, so 0.3F≥27.40.3F \geq 27.4, thus F≥91.33F \geq 91.33. Since exam scores must be whole numbers, the minimum needed is 9292 points. Checking: with F=92F = 92, the final grade is 0.7(78)+0.3(92)=54.6+27.6=82.2≥820.7(78) + 0.3(92) = 54.6 + 27.6 = 82.2 \geq 82. With F=91F = 91, the final grade would be 54.6+27.3=81.9<8254.6 + 27.3 = 81.9 < 82, which is insufficient.

Question 4

A student has 68 points in a class. Each practice quiz adds 2 points. The student needs more than 75 points to earn a B. Let qq be the number of quizzes. Solve the inequality and choose the correct statement about how many whole quizzes are needed.

  1. 68+2q>75⇒2q>7⇒q>768+2q>75\Rightarrow 2q>7\Rightarrow q>7, so q≥8q\ge 8 whole quizzes
  2. 68+2q≥75⇒2q≥7⇒q≥3.568+2q\ge 75\Rightarrow 2q\ge 7\Rightarrow q\ge 3.5, so q≥3q\ge 3 whole quizzes
  3. 68+2q>75⇒2q>7⇒q>3.568+2q>75\Rightarrow 2q>7\Rightarrow q>3.5, so q≥4q\ge 4 whole quizzes (correct answer)
  4. 68+2q>75⇒q>3.568+2q>75\Rightarrow q>3.5, so q=3.5q=3.5 quizzes is enough
Explanation: This question tests solving two-step linear inequalities from word problems, graphing solution sets on number lines (open/closed dots, directional shading), and interpreting in context, especially with whole numbers. Solving is similar to equations but preserves inequality direction—for px+q≤r: subtract q (px≤r-q), divide by p (x≤(r-q)/p if p>0, flip to x≥... if p<0); graphing uses closed dot ● for ≤ or ≥ (includes boundary), open ○ for < or > (excludes), shade left for < or ≤ (toward smaller), right for > or ≥ (toward larger); example: 5v+10≤100 → 5v≤90 → v≤18 graphs as closed dot at 18, shaded left. For needing more than 75 points with 68 current and +2 per quiz, set up 68+2q>75, solve by subtracting 68 (2q>7), dividing by 2 (q>3.5), and since quizzes are whole, q≥4 as q=4 gives 76>75 but q=3 gives 74≤75. The correct choice A reflects this with q>3.5 so q≥4 whole quizzes. Errors include using ≥75 leading to q≥3 incorrectly (choice B), dividing wrong to q>7 (choice C), or ignoring whole numbers by suggesting q=3.5 (choice D). Steps are translating 'more than' to >75, solving with inverse operations keeping direction, interpreting for whole q≥4 since context requires integers (can't do half quizzes, round up). Mistakes involve direction errors like using ≥ instead of >, forgetting to round up for strict inequality, or calculation slips like dividing 7 by 2 incorrectly.

Question 5

A phone plan charges a $12 monthly fee plus $0.25 per text message. You want your bill to be less than $20.

Set up, solve, and graph an inequality for the number of texts $x$ you can send.

  1. 0.25x+12<20⇒x<80.25x+12<20 \Rightarrow x<8; graph: open dot at 8 and shade left
  2. 0.25x+12<20⇒0.25x<8⇒x<320.25x+12<20 \Rightarrow 0.25x<8 \Rightarrow x<32; graph: open dot at 32 and shade left (correct answer)
  3. 0.25x+12≤20⇒x≤320.25x+12\le 20 \Rightarrow x\le 32; graph: closed dot at 32 and shade left
  4. 0.25x+12<20⇒x>320.25x+12<20 \Rightarrow x>32; graph: open dot at 32 and shade right
Explanation: This question tests solving two-step linear inequalities from word problems, graphing solution sets on number lines (open/closed dots, directional shading), and interpreting in context. Solving is similar to equations but preserves inequality direction—for px+q≤r: subtract q (px≤r-q), divide by p (x≤(r-q)/p if p>0, flip to x≥... if p<0); graphing uses closed dot ● for ≤ or ≥ (includes boundary), open ○ for < or > (excludes), shade left for < or ≤ (toward smaller), right for > or ≥ (toward larger); example: 5v+10≤100 → 5v≤90 → v≤18 graphs as closed dot at 18, shaded left. For this phone plan with $12 fee plus $0.25 per text and less than $20 bill, set up 0.25x+12<20, solve by subtracting 12 (0.25x<8), dividing by 0.25 (x<32), graph open dot at 32 shaded left, interpreting as fewer than 32 texts. The correct choice is A: 0.25x+12<20 ⇒ x<32, graph open dot at 32 and shade left. Common errors include dividing incorrectly like x<8 (choice B), using ≤ and closed dot (choice C), or flipping to > (choice D). Steps: (1) translate 'less than $20' to <20, (2) solve with inverse operations, (3) keep direction since dividing by positive 0.25, (4) graph with open dot and left shade, (5) interpret x<32 as 31,30,...,0 texts possible. Mistakes often involve boundary type wrong or shading the wrong way.

Question 6

A video game download is 1.5 GB plus 0.5 GB for each extra level pack you add. You have less than 5 GB of free space.

Set up, solve, and graph an inequality for the number of level packs pp you can add.

  1. 0.5p+1.5<5⇒0.5p<3.5⇒p<70.5p+1.5<5 \Rightarrow 0.5p<3.5 \Rightarrow p<7; graph: open dot at 7 and shade left (correct answer)
  2. 0.5p+1.5≤5⇒p≤70.5p+1.5\le 5 \Rightarrow p\le 7; graph: closed dot at 7 and shade left
  3. 0.5p+1.5<5⇒p>70.5p+1.5<5 \Rightarrow p>7; graph: open dot at 7 and shade right
  4. 0.5p+1.5<5⇒p<3.50.5p+1.5<5 \Rightarrow p<3.5; graph: open dot at 3.5 and shade left
Explanation: This question tests solving two-step linear inequalities from word problems, graphing solution sets on number lines (open/closed dots, directional shading), and interpreting in context. Solving is similar to equations but preserves inequality direction—for px+q≤r: subtract q (px≤r-q), divide by p (x≤(r-q)/p if p>0, flip to x≥... if p<0); graphing uses closed dot ● for ≤ or ≥ (includes boundary), open ○ for < or > (excludes), shade left for < or ≤ (toward smaller), right for > or ≥ (toward larger); example: 5v+10≤100 → 5v≤90 → v≤18 graphs as closed dot at 18, shaded left. For this game download 1.5 GB plus 0.5 GB per pack with less than 5 GB space, set up 0.5p+1.5<5, solve by subtracting 1.5 (0.5p<3.5), dividing by 0.5 (p<7), graph open dot at 7 shaded left, interpreting as fewer than 7 packs. The correct choice is A: 0.5p+1.5<5 ⇒ p<7, graph open dot at 7 and shade left. Common errors include dividing incorrectly like p<3.5 (choice B), using ≤ and closed dot (choice C), or flipping to > (choice D). Steps: (1) translate 'less than 5' to <5, (2) solve with inverse operations, (3) keep direction since dividing by positive 0.5, (4) graph with open dot and left shade, (5) interpret p<7 as 6,5,...,0 packs. Mistakes often involve calculation errors or shading wrong direction.

Question 7

To pass a class, your score must be more than 75 points. You have 68 points now. Each completed practice quiz adds 2 points. Let qq be the number of quizzes.

Solve the inequality and state the smallest whole number of quizzes you must complete.

Inequality: 68+2q>7568 + 2q > 75

  1. q>3.5q>3.5, so you need at least 44 quizzes. (correct answer)
  2. q≥3.5q\ge3.5, so you need at least 3.53.5 quizzes.
  3. q>3.5q>3.5, so you need at least 33 quizzes.
  4. q≥4q\ge4, so you need exactly 44 quizzes.
Explanation: This question tests solving two-step linear inequalities from word problems, graphing solution sets on number lines (open/closed dots, directional shading), and interpreting in context with whole numbers. Solving is similar to equations but preserves inequality direction—for px+q≤r: subtract q (px≤r-q), divide by p (x≤(r-q)/p if p>0, flip to x≥... if p<0); graphing uses closed dot ● for ≤ or ≥ (includes boundary), open ○ for < or > (excludes), shade left for < or ≤ (toward smaller), right for > or ≥ (toward larger); example: 5v+10≤100 → 5v≤90 → v≤18 graphs as closed dot at 18, shaded left for 18 or less. For this quiz problem needing more than 75 points from 68 plus 2 per quiz, set up 68+2q>75, solve by subtracting 68 (2q>7), dividing by 2 (q>3.5 since 2>0, no flip), and interpret smallest whole number as 4 quizzes since q must be integer. The correct choice A shows q>3.5 and at least 4 quizzes. Errors include using ≥ instead of > (choice C with q≥3.5), rounding down incorrectly (choice B with at least 3), or misstating as exactly 4 (choice D). Steps include: (1) translate 'more than 75' to >75, (2) solve with inverse operations, (3) for strict >, use open circle if graphing, (4) interpret with ceiling to next integer for 'at least'. Context requires whole quizzes, and mistakes involve boundary type or forgetting to round up for strict inequality.

Question 8

A temperature starts at −5∘C-5^\circ\text{C} and rises 3∘C3^\circ\text{C} per hour. After how many hours will the temperature be greater than 10∘C10^\circ\text{C}?

Solve and graph the solution set on a number line.

Inequality: −5+3h>10-5 + 3h > 10​​

  1. h≥5h\ge5. Graph: closed dot at 5, shade right. Interpretation: after 5 hours or more.
  2. h>53h>\frac{5}{3}. Graph: open circle at 53\frac{5}{3}, shade right. Interpretation: after more than 53\frac{5}{3} hours.
  3. h>5h>5. Graph: open circle at 5, shade right. Interpretation: after more than 5 hours. (correct answer)
  4. h<5h<5. Graph: open circle at 5, shade left. Interpretation: before 5 hours.
Explanation: This question tests solving two-step linear inequalities from word problems, graphing solution sets on number lines (open/closed dots, directional shading), and interpreting in context. Solving is similar to equations but preserves inequality direction—for px+q≤r: subtract q (px≤r-q), divide by p (x≤(r-q)/p if p>0, flip to x≥... if p<0); graphing uses closed dot ● for ≤ or ≥ (includes boundary), open ○ for < or > (excludes), shade left for < or ≤ (toward smaller), right for > or ≥ (toward larger); example: 5v+10≤100 → 5v≤90 → v≤18 graphs as closed dot at 18, shaded left for 18 or less. For this temperature problem starting at -5°C rising 3°C per hour to greater than 10°C, set up -5+3h>10, solve by adding 5 (3h>15), dividing by 3 (h>5), graph with open circle at 5 shaded right, and interpret as after more than 5 hours. The correct choice A shows h>5, open circle at 5 shaded right, and interpretation after more than 5 hours. Errors include using ≥ (choice B with closed dot), wrong direction (choice C with h<5 shaded left), or arithmetic error (choice D with h>5/3). Steps include: (1) translate 'greater than' to >, (2) solve by adding and dividing without flipping, (3) graph with open circle and right shading for >, (4) interpret as h>5 meaning more than 5. Mistakes often involve boundary type wrong or shading the wrong direction.

Question 9

A movie theater sells a ticket for $12 and charges a $3 online fee per order. You have $30 and want to spend at most $30 total. Let $t$ be the number of tickets.

Solve and graph: 12t+3≤3012t + 3 \le 30

  1. t<2.25t<2.25. Graph: open circle at 2.25, shade left. Interpretation: at most 2 tickets (whole tickets).
  2. t≤2.25t\le2.25. Graph: closed dot at 2.25, shade left. Interpretation: at most 2 tickets (whole tickets). (correct answer)
  3. t≤2.75t\le2.75. Graph: closed dot at 2.75, shade left. Interpretation: at most 2 tickets (whole tickets).
  4. t≤2.25t\le2.25. Graph: closed dot at 2.25, shade left. Interpretation: up to 2.25 tickets.
Explanation: This question tests solving two-step linear inequalities from word problems, graphing solution sets on number lines (open/closed dots, directional shading), and interpreting in context with whole numbers. Solving is similar to equations but preserves inequality direction—for px+q≤r: subtract q (px≤r-q), divide by p (x≤(r-q)/p if p>0, flip to x≥... if p<0); graphing uses closed dot ● for ≤ or ≥ (includes boundary), open ○ for < or > (excludes), shade left for < or ≤ (toward smaller), right for > or ≥ (toward larger); example: 5v+10≤100 → 5v≤90 → v≤18 graphs as closed dot at 18, shaded left for 18 or less. For this movie ticket problem with $12 per ticket plus $3 fee, at most $30, set up 12t+3≤30, solve subtracting 3 (12t≤27), dividing by 12 (t≤2.25), graph closed dot at 2.25 shaded left, and interpret as at most 2 whole tickets. The correct choice B shows t≤2.25 graph and interpretation at most 2 tickets. Errors include literal without context (choice A), miscalculation (choice C with t≤2.75), or strict < (choice D with open circle). Steps include: (1) translate 'at most' to ≤, (2) solve subtracting and dividing no flip, (3) graph closed left for ≤, (4) interpret with floor to 2 for whole tickets. Mistakes involve ignoring context for integers or boundary type.

Question 10

A streaming service charges a $4 sign-up fee plus $6 per month. You want to spend less than $40 total. Let $m$ be the number of months.

Solve the inequality and graph the solution set on a number line.

Inequality: 6m+4<406m + 4 < 40

  1. m<346m<\frac{34}{6}. Graph: open circle at 346\frac{34}{6}, shade right. Interpretation: more than 346\frac{34}{6} months.
  2. m<406m<\frac{40}{6}. Graph: open circle at 406\frac{40}{6}, shade left. Interpretation: fewer than 406\frac{40}{6} months.
  3. m<6m<6. Graph: open circle at 6, shade left. Interpretation: fewer than 6 months. (correct answer)
  4. m≤6m\le6. Graph: closed dot at 6, shade left. Interpretation: at most 6 months.
Explanation: This question tests solving two-step linear inequalities from word problems, graphing solution sets on number lines (open/closed dots, directional shading), and interpreting in context. Solving is similar to equations but preserves inequality direction—for px+q≤r: subtract q (px≤r-q), divide by p (x≤(r-q)/p if p>0, flip to x≥... if p<0); graphing uses closed dot ● for ≤ or ≥ (includes boundary), open ○ for < or > (excludes), shade left for < or ≤ (toward smaller), right for > or ≥ (toward larger); example: 5v+10≤100 → 5v≤90 → v≤18 graphs as closed dot at 18, shaded left for 18 or less. For this streaming service with $4 fee plus $6 per month, less than $40 total, set up 6m+4<40, solve by subtracting 4 (6m<36), dividing by 6 (m<6), graph with open circle at 6 shaded left, and interpret as fewer than 6 months. The correct choice A shows m<6, open circle at 6 shaded left, and fewer than 6 months. Errors include using ≤ (choice B with closed dot), arithmetic slips (choice C with m<40/6), or wrong shading/direction (choice D with m<34/6 shaded right). Steps include: (1) translate 'less than' to <, (2) solve with subtraction and division, no flip, (3) graph open circle left for <, (4) interpret as m<6. Context might limit to positive months, and common mistakes are boundary type or incorrect subtraction.

Question 11

A phone plan starts with $50 credit, and you are charged $2.50 per day you use data. You want your remaining credit to be at least $20. Let $d$ be the number of days you use data.

Set up and solve the inequality: 50−2.5d≥2050 - 2.5d \ge 20

  1. d≤12d\le12. Interpretation: you can use data for 12 days or fewer. (correct answer)
  2. d≤30d\le30. Interpretation: you can use data for 30 days or fewer.
  3. d≥12d\ge12. Interpretation: you must use data for at least 12 days.
  4. d≥30d\ge30. Interpretation: you must use data for at least 30 days.
Explanation: This question tests solving two-step linear inequalities from word problems, graphing solution sets on number lines (open/closed dots, directional shading), and interpreting in context. Solving is similar to equations but preserves inequality direction—for px+q≤r: subtract q (px≤r-q), divide by p (x≤(r-q)/p if p>0, flip to x≥... if p<0); graphing uses closed dot ● for ≤ or ≥ (includes boundary), open ○ for < or > (excludes), shade left for < or ≤ (toward smaller), right for > or ≥ (toward larger); example: 5v+10≤100 → 5v≤90 → v≤18 graphs as closed dot at 18, shaded left for 18 or less. For this phone plan starting at $50 credit minus $2.50 per data day, at least $20 remaining, set up 50-2.5d≥20, solve by subtracting 50 (-2.5d≥-30), dividing by -2.5 (d≤12, flip since negative), and interpret as 12 or fewer days. The correct choice A shows d≤12 and interpretation of 12 days or fewer. Errors include no flip (choice B with d≥12), wrong constants (choices C and D with d≤30 or d≥30). Steps include: (1) translate 'at least $20' to ≥20, (2) solve by subtracting and dividing with flip for negative, (3) graph would be closed dot at 12 shaded left, (4) interpret as d≤12. Mistakes often involve forgetting to flip when dividing by negative or direction errors.

Question 12

A streaming service charges $8 per month plus $2 per movie rented. You want to spend less than $20 this month. Let $m$ be the number of movies. Solve the inequality and graph the solution on a number line.

  1. 2m+8<20⇒2m<12⇒m<62m+8<20\Rightarrow 2m<12\Rightarrow m<6; graph: open dot at 6, shade left (correct answer)
  2. 2m+8<20⇒m<122m+8<20\Rightarrow m<12; graph: open dot at 12, shade left
  3. 2m+8<20⇒2m<28⇒m<142m+8<20\Rightarrow 2m<28\Rightarrow m<14; graph: open dot at 14, shade left
  4. 2m+8≤20⇒2m≤12⇒m≤62m+8\le 20\Rightarrow 2m\le 12\Rightarrow m\le 6; graph: closed dot at 6, shade left
Explanation: This question tests solving two-step linear inequalities from word problems, graphing solution sets on number lines (open/closed dots, directional shading), and interpreting in context. Solving is similar to equations but preserves inequality direction—for px+q≤r: subtract q (px≤r-q), divide by p (x≤(r-q)/p if p>0, flip to x≥... if p<0); graphing uses closed dot ● for ≤ or ≥ (includes boundary), open ○ for < or > (excludes), shade left for < or ≤ (toward smaller), right for > or ≥ (toward larger); example: 5v+10≤100 → 5v≤90 → v≤18 graphs as closed dot at 18, shaded left. For $8 monthly plus $2 per movie, less than $20, set up 2m+8<20, subtract 8 (2m<12), divide by 2 (m<6), graph open dot at 6 shaded left, meaning fewer than 6 movies. The correct choice A reflects this solution and graph. Errors include using ≤ with closed dot (choice B), forgetting to divide by 2 (choice C), or subtracting wrong to m<14 (choice D). Steps include translating 'less than' to <20, solving with inverse operations keeping direction, graphing open and left, interpreting m<6 (like 0 to 5 movies). Mistakes often involve changing < to ≤, shading wrong direction, or one-step errors like not dividing.

Question 13

A gym charges a $10 monthly base fee plus $5 per visit. The gym requires at least 12 visits per month to keep a special membership. You want to spend no more than $100 this month. Write, solve, and graph an inequality for the number of visits v that meet both conditions.

  1. 5v+10>=100 and v>=12; from the cost: v>=18; combined: v>=18; graph: closed dot at 18 and shade right
  2. 5v+10<=100 and v>=12; from the cost: v<=90; combined: 12<=v<=90; graph: closed dots at 12 and 90 with shading between
  3. 5v+10<=100 and v>=12; from the cost: v<=18; combined: 12<=v<=18; graph: closed dots at 12 and 18 with shading between (correct answer)
  4. 5v+10<=100 and v<=12; from the cost: v<=18; combined: v<=12; graph: closed dot at 12 and shade left
Explanation: The budget condition is 5v + 10 <= 100, which solves to 5v <= 90 and then v <= 18. Combined with the membership requirement v >= 12, the solution is 12 <= v <= 18, graphed as closed dots at 12 and 18 with shading between them, matching choice C. Choice A incorrectly uses >= 100 for the budget condition, which doesn't match 'no more than $100.' Choice B skips dividing by 5, leaving v <= 90 instead of v <= 18. Choice D reverses the membership condition to v <= 12, combining incorrectly.

Question 14

A student has xx minutes of free time. They spend 15 minutes on homework and then want to have at least 25 minutes left to play outside.

Which inequality correctly represents this situation, and what is the solution?

(Assume xx is the total free time in minutes.)

  1. Inequality: x−15≥25x-15\ge25. Solution: x≥40x\ge40. Graph: closed dot at 40, shade right. (correct answer)
  2. Inequality: 15−x≥2515-x\ge25. Solution: x≤−10x\le-10. Graph: closed dot at -10, shade left.
  3. Inequality: x−15≤25x-15\le25. Solution: x≤40x\le40. Graph: closed dot at 40, shade left.
  4. Inequality: x+15≥25x+15\ge25. Solution: x≥10x\ge10. Graph: closed dot at 10, shade right.
Explanation: This question tests solving two-step linear inequalities from word problems, graphing solution sets on number lines (open/closed dots, directional shading), and interpreting in context. Solving is similar to equations but preserves inequality direction—for px+q≤r: subtract q (px≤r-q), divide by p (x≤(r-q)/p if p>0, flip to x≥... if p<0); graphing uses closed dot ● for ≤ or ≥ (includes boundary), open ○ for < or > (excludes), shade left for < or ≤ (toward smaller), right for > or ≥ (toward larger); example: 5v+10≤100 → 5v≤90 → v≤18 graphs as closed dot at 18, shaded left for 18 or less. For this free time problem with x minutes, minus 15 for homework, at least 25 left, the correct inequality is x-15≥25, solving to x≥40, graphed closed dot at 40 shaded right. The correct choice A shows x-15≥25, x≥40, closed dot at 40 shaded right. Errors include wrong direction (choice B with ≤40 shaded left), reversed subtraction (choice C with 15-x≥25 to x≤-10), or addition (choice D with x+15≥25 to x≥10). Steps include: (1) translate 'at least 25 left' to remaining ≥25, so x-15≥25, (2) solve adding 15, no flip, (3) graph closed right for ≥, (4) interpret as x≥40. Common mistakes are inequality setup direction or confusing subtraction with addition.

Question 15

A student has xx minutes of free time. They spend 15 minutes on homework and then want to have at least 25 minutes left to play outside. Which inequality correctly represents this situation, and what is the solution? (Assume xx is the total free time in minutes.)

  1. Inequality: 15−x≥2515-x\ge25. Solution: x≤−10x\le-10. Graph: closed dot at -10, shade left.
  2. Inequality: x−15≤25x-15\le25. Solution: x≤40x\le40. Graph: closed dot at 40, shade left.
  3. Inequality: x−15≥25x-15\ge25. Solution: x≥40x\ge40. Graph: closed dot at 40, shade right. (correct answer)
  4. Inequality: x+15≥25x+15\ge25. Solution: x≥10x\ge10. Graph: closed dot at 10, shade right.
Explanation: Starting with x minutes of free time and subtracting the 15 minutes spent on homework must leave at least 25 minutes, so the correct inequality is x minus 15 is greater than or equal to 25, which solves to x is greater than or equal to 40, matching choice C. Choice A reverses the subtraction order to 15 minus x, giving an inequality that does not represent the situation. Choice B uses the wrong inequality direction, less than or equal to, instead of at least. Choice D incorrectly adds 15 instead of subtracting it. Since the solution is x is greater than or equal to 40, the graph needs a closed dot at 40 with shading to the right.

Question 16

A movie theater sells a ticket for $12 and charges a $3 online fee per order. You have $30 and want to spend at most $30 total. Let $t$ be the number of tickets.

Solve and graph: 12t+3≤3012t + 3 \le 30​​

  1. t≤2.25t\le2.25. Graph: closed dot at 2.25, shade left. Interpretation: at most 2 tickets (whole tickets). (correct answer)
  2. t≤2.75t\le2.75. Graph: closed dot at 2.75, shade left. Interpretation: at most 2 tickets (whole tickets).
  3. t<2.25t<2.25. Graph: open circle at 2.25, shade left. Interpretation: at most 2 tickets (whole tickets).
  4. t≤2.25t\le2.25. Graph: closed dot at 2.25, shade left. Interpretation: up to 2.25 tickets.
Explanation: This question tests solving two-step linear inequalities from word problems, graphing solution sets on number lines (open/closed dots, directional shading), and interpreting in context with whole numbers. Solving is similar to equations but preserves inequality direction—for px+q≤r: subtract q (px≤r-q), divide by p (x≤(r-q)/p if p>0, flip to x≥... if p<0); graphing uses closed dot ● for ≤ or ≥ (includes boundary), open ○ for < or > (excludes), shade left for < or ≤ (toward smaller), right for > or ≥ (toward larger); example: 5v+10≤100 → 5v≤90 → v≤18 graphs as closed dot at 18, shaded left for 18 or less. For this movie ticket problem with $12 per ticket plus $3 fee, at most $30, set up 12t+3≤30, solve subtracting 3 (12t≤27), dividing by 12 (t≤2.25), graph closed dot at 2.25 shaded left, and interpret as at most 2 whole tickets. The correct choice B shows t≤2.25 graph and interpretation at most 2 tickets. Errors include literal without context (choice A), miscalculation (choice C with t≤2.75), or strict < (choice D with open circle). Steps include: (1) translate 'at most' to ≤, (2) solve subtracting and dividing no flip, (3) graph closed left for ≤, (4) interpret with floor to 2 for whole tickets. Mistakes involve ignoring context for integers or boundary type.

Question 17

The solution to the inequality 2x−63≤4\frac{2x - 6}{3} \leq 4 is graphed on the number line. Which value is NOT in the solution set?

  1. x=9x = 9
  2. x=7x = 7
  3. x=8x = 8
  4. x=10x = 10 (correct answer)
Explanation: Solving 2x−63≤4\frac{2x - 6}{3} \leq 4: multiply by 33 to get 2x−6≤122x - 6 \leq 12, so 2x≤182x \leq 18, thus x≤9x \leq 9. The graph shows a closed circle at 99 with shading to the left. Values 77, 88, and 99 are all ≤9\leq 9, so they're in the solution set. However, x=10>9x = 10 > 9, so it's not in the solution set.

Question 18

The number line represents the solution to the inequality −2x+7>1-2x + 7 > 1. Which statement about the solution is correct?

  1. All values greater than 33 are solutions to the inequality
  2. All values less than 33 are solutions to the inequality (correct answer)
  3. The value x=3x = 3 is included in the solution set
  4. All values greater than or equal to 33 are solutions
Explanation: Solving −2x+7>1-2x + 7 > 1: −2x>−6-2x > -6, so x<3x < 3 (inequality flips when dividing by negative number). The graph shows an open circle at 33 with shading to the left, confirming x<3x < 3. Choice A incorrectly states x>3x > 3. Choice C is wrong because the open circle means x=3x = 3 is not included. Choice D incorrectly includes x=3x = 3 and has the wrong direction.

Question 19

A streaming service charges a $4 sign-up fee plus $6 per month. You want to spend less than $40 total. Let $mbethenumberofmonths.Solvetheinequalityandgraphthesolutionsetonanumberline.Inequality:be the number of months. Solve the inequality and graph the solution set on a number line. Inequality:6m + 4 < 40$

  1. m<6m<6. Graph: open circle at 6, shade left. Interpretation: fewer than 6 months. (correct answer)
  2. m<346m<\frac{34}{6}. Graph: open circle at 346\frac{34}{6}, shade right. Interpretation: more than 346\frac{34}{6} months.
  3. m<406m<\frac{40}{6}. Graph: open circle at 406\frac{40}{6}, shade left. Interpretation: fewer than 406\frac{40}{6} months.
  4. m≤6m\le6. Graph: closed dot at 6, shade left. Interpretation: at most 6 months.
Explanation: Subtract 4 from both sides: 6m<36. Divide by 6: m<6. Since the inequality is strict, use an open circle at 6 shaded left, meaning fewer than 6 months, matching choice A. Choice B makes an arithmetic and direction error, giving m<34/6 with shading to the right. Choice C makes an arithmetic slip when isolating m, giving m<40/6 instead of m<6. Choice D uses a closed dot instead of an open circle, which would only be correct for a non-strict inequality.

Question 20

A gym charges a $10 monthly base fee plus $5 per visit. You want to spend at most $100 this month. If $vrepresentsthenumberofvisits,whichofthefollowingcorrectlysolvestheinequalityrepresents the number of visits, which of the following correctly solves the inequality5v + 10 \le 100$ and correctly describes the graph of the solution?

  1. Solve: 5v+10≤100⇒5v≤110⇒v≤225v+10\le100\Rightarrow 5v\le110\Rightarrow v\le22. Graph: closed dot at 22, shade left. Interpretation: 22 or fewer visits.
  2. Solve: 5v+10≤100⇒5v≤90⇒v≤185v+10\le100\Rightarrow 5v\le90\Rightarrow v\le18. Graph: closed dot at 18, shade left. Interpretation: 18 or fewer visits. (correct answer)
  3. Solve: 5v+10≤100⇒v≤905v+10\le100\Rightarrow v\le90. Graph: closed dot at 90, shade left. Interpretation: 90 or fewer visits.
  4. Solve: 5v+10≤100⇒5v≤90⇒v≥185v+10\le100\Rightarrow 5v\le90\Rightarrow v\ge18. Graph: closed dot at 18, shade right. Interpretation: 18 or more visits.
Explanation: Subtracting 10 from both sides gives 5v <= 90, and dividing both sides by 5 (a positive number, so the inequality sign stays the same) gives v <= 18. On a number line, this is shown with a closed dot at 18 and shading to the left, meaning 18 or fewer visits keeps spending at or under $100. Choice A is wrong because it adds 10 instead of subtracting it. Choice C is wrong because it skips dividing by 5 after subtracting. Choice D is wrong because it flips the inequality sign, which should only happen when dividing by a negative number.