7th Grade Math Quiz: Solve And Graph Linear Inequalities

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 7th Grade 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.

Question 1

1. $6$ miles
2. $7$ miles
3. $8$ miles (correct answer)
4. $9$ miles

Explanation: For $m$ miles where $m > 1$, the cost is $3.50 + 2.25(m-1)$. The inequality is $3.50 + 2.25(m-1) \leq 20$. Expanding: $3.50 + 2.25m - 2.25 \leq 20$, so $1.25 + 2.25m \leq 20$, giving $2.25m \leq 18.75$, thus $m \leq 8.33$. Since we need whole miles, the maximum is $8$ miles. Checking: $8$ miles costs 3.50 + 2.25(7) = \19.25 \leq $20$. For$9$miles:$3.50 + 2.25(8) = $21.50 > $20$, which exceeds the budget.

Question 2

1. Chen can send at most $300$ text messages per month (correct answer)
2. Chen can send at most $250$ text messages per month
3. Chen can send at most $200$ text messages per month
4. Chen can send at most $350$ text messages per month

Explanation: If Chen sends $t$ messages where $t > 200$, his bill is $40 + 0.15(t - 200)$. The inequality is $40 + 0.15(t - 200) \leq 55$, giving $0.15(t - 200) \leq 15$, so $t - 200 \leq 100$, thus $t \leq 300$. Choice B ($250$) is unnecessarily restrictive. Choice C ($200$) ignores that he can go over the base amount. Choice D ($350$) would cost $40 + 0.15(150) = \$62.50$, exceeding his budget.

Question 3

1. The student needs at least $88$ points on the final exam
2. The student needs at least $90$ points on the final exam
3. The student needs at least $92$ points on the final exam (correct answer)
4. The student needs at least $94$ points on the final exam

Explanation: Setting up the inequality: $0.7(78) + 0.3F \geq 82$. This gives $54.6 + 0.3F \geq 82$, so $0.3F \geq 27.4$, thus $F \geq 91.33$. Since exam scores must be whole numbers, the minimum needed is $92$ points. Checking: with $F = 92$, the final grade is $0.7(78) + 0.3(92) = 54.6 + 27.6 = 82.2 \geq 82$. With $F = 91$, the final grade would be $54.6 + 27.3 = 81.9 < 82$, which is insufficient.

Question 4

1. $68+2q>75\Rightarrow 2q>7\Rightarrow q>7$, so $q\ge 8$ whole quizzes
2. $68+2q\ge 75\Rightarrow 2q\ge 7\Rightarrow q\ge 3.5$, so $q\ge 3$ whole quizzes
3. $68+2q>75\Rightarrow 2q>7\Rightarrow q>3.5$, so $q\ge 4$ whole quizzes (correct answer)
4. $68+2q>75\Rightarrow q>3.5$, so $q=3.5$ 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

1. $0.25x+12<20 \Rightarrow x<8$; graph: open dot at 8 and shade left
2. $0.25x+12<20 \Rightarrow 0.25x<8 \Rightarrow x<32$; graph: open dot at 32 and shade left (correct answer)
3. $0.25x+12\le 20 \Rightarrow x\le 32$; graph: closed dot at 32 and shade left
4. $0.25x+12<20 \Rightarrow x>32$; 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

1. $0.5p+1.5<5 \Rightarrow 0.5p<3.5 \Rightarrow p<7$; graph: open dot at 7 and shade left (correct answer)
2. $0.5p+1.5\le 5 \Rightarrow p\le 7$; graph: closed dot at 7 and shade left
3. $0.5p+1.5<5 \Rightarrow p>7$; graph: open dot at 7 and shade right
4. $0.5p+1.5<5 \Rightarrow p<3.5$; 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

1. $q>3.5$, so you need at least $4$ quizzes. (correct answer)
2. $q\ge3.5$, so you need at least $3.5$ quizzes.
3. $q>3.5$, so you need at least $3$ quizzes.
4. $q\ge4$, so you need exactly $4$ 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

1. $h\ge5$. Graph: closed dot at 5, shade right. Interpretation: after 5 hours or more.
2. $h>\frac{5}{3}$. Graph: open circle at $\frac{5}{3}$, shade right. Interpretation: after more than $\frac{5}{3}$ hours.
3. $h>5$. Graph: open circle at 5, shade right. Interpretation: after more than 5 hours. (correct answer)
4. $h<5$. 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

1. $5v+10\le 60\Rightarrow 5v\le 50\Rightarrow v\le 10$; graph: closed dot at 10, shade left (correct answer)
2. $5v+10<60\Rightarrow 5v<50\Rightarrow v<10$; graph: open dot at 10, shade left
3. $5v+10\ge 60\Rightarrow 5v\ge 50\Rightarrow v\ge 10$; graph: closed dot at 10, shade right
4. $5v+10\le 60\Rightarrow v\le 50$; graph: closed dot at 50, 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 gym problem with $10 base plus$5 per visit and at most $60, set up 5v+10≤60, solve by subtracting 10 (5v≤50), dividing by 5 (v≤10), and graph with closed dot at 10 shaded left, meaning 10 or fewer visits stay within budget. The correct choice A shows this solution and graph accurately. Common errors include reversing the inequality to ≥ (choice B), forgetting to divide by 5 after subtracting (choice C), or using strict < with open dot (choice D). Steps include translating 'at most' to ≤60, solving with inverse operations while keeping direction since dividing by positive 5, graphing with closed dot and left shading, and interpreting as v≤10 for budget compliance. Mistakes often involve wrong inequality direction, shading right instead of left, or incorrect boundary type like open for ≤.

Question 10

1. $t<2.25$. Graph: open circle at 2.25, shade left. Interpretation: at most 2 tickets (whole tickets).
2. $t\le2.25$. Graph: closed dot at 2.25, shade left. Interpretation: at most 2 tickets (whole tickets). (correct answer)
3. $t\le2.75$. Graph: closed dot at 2.75, shade left. Interpretation: at most 2 tickets (whole tickets).
4. $t\le2.25$. 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 11

1. $m<\frac{34}{6}$. Graph: open circle at $\frac{34}{6}$, shade right. Interpretation: more than $\frac{34}{6}$ months.
2. $m<\frac{40}{6}$. Graph: open circle at $\frac{40}{6}$, shade left. Interpretation: fewer than $\frac{40}{6}$ months.
3. $m<6$. Graph: open circle at 6, shade left. Interpretation: fewer than 6 months. (correct answer)
4. $m\le6$. 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 12

1. $d\le12$. Interpretation: you can use data for 12 days or fewer. (correct answer)
2. $d\le30$. Interpretation: you can use data for 30 days or fewer.
3. $d\ge12$. Interpretation: you must use data for at least 12 days.
4. $d\ge30$. 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 13

1. $2m+8<20\Rightarrow 2m<12\Rightarrow m<6$; graph: open dot at 6, shade left (correct answer)
2. $2m+8<20\Rightarrow m<12$; graph: open dot at 12, shade left
3. $2m+8<20\Rightarrow 2m<28\Rightarrow m<14$; graph: open dot at 14, shade left
4. $2m+8\le 20\Rightarrow 2m\le 12\Rightarrow m\le 6$; 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 14

1. -4x+6>=18 => -4x>=12 => x<=-3; graph: closed dot at -3, shade left (correct answer)
2. -4x+6>=18 => -4x>=24 => x<=-6; graph: closed dot at -6, shade left
3. -4x+6>=18 => x<=3; graph: closed dot at 3, shade left
4. -4x+6>=18 => -4x>=12 => x>=-3; graph: closed dot at -3, shade right

Explanation: Subtracting 6 from both sides gives -4x >= 12. Dividing by -4 flips the inequality, giving x <= -3, graphed as a closed dot at -3 with shading to the left, matching choice A. Choice D keeps the correct numbers but forgets to flip the inequality, giving the wrong direction (x >= -3, shaded right). Choice B makes an arithmetic error subtracting 6 from 18, and choice C skips the division step entirely, landing on the wrong value with no flip.

Question 15

1. $-3x+9\ge 0 \Rightarrow -3x\ge -9 \Rightarrow x\le 3$; graph: closed dot at 3 and shade left (correct answer)
2. $-3x+9\ge 0 \Rightarrow -3x\ge -9 \Rightarrow x\ge 3$; graph: closed dot at 3 and shade right
3. $-3x+9\ge 0 \Rightarrow x\le -3$; graph: closed dot at -3 and shade left
4. $-3x+9\ge 0 \Rightarrow -3x\le -9 \Rightarrow x\le 3$; graph: closed dot at 3 and shade left

Explanation: This question tests solving two-step linear inequalities, graphing solution sets on number lines (open/closed dots, directional shading), and handling negative coefficients. 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 -3x+9≥0, solve by subtracting 9 (-3x≥-9), dividing by -3 and flipping (x≤3), graph closed dot at 3 shaded left. The correct choice is A: -3x+9≥0 ⇒ x≤3, graph closed dot at 3 and shade left. Common errors include not flipping when dividing by negative (choice B), incorrect intermediate step like ≤ instead of ≥ (choice C), or wrong boundary like x≤-3 (choice D). Steps: (1) start with given inequality, (2) solve with inverse operations, (3) flip direction when dividing by negative -3, (4) graph with closed dot and left shade, (5) interpret x≤3 as 3,2,1,... down to negative infinity. Mistakes often involve forgetting to flip the inequality or shading the wrong way.

Question 16

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 17

1. Inequality: $x-15\ge25$. Solution: $x\ge40$. Graph: closed dot at 40, shade right. (correct answer)
2. Inequality: $15-x\ge25$. Solution: $x\le-10$. Graph: closed dot at -10, shade left.
3. Inequality: $x-15\le25$. Solution: $x\le40$. Graph: closed dot at 40, shade left.
4. Inequality: $x+15\ge25$. Solution: $x\ge10$. 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 18

1. Solve: $5v+10\le100\Rightarrow 5v\le90\Rightarrow v\le18$. Graph: closed dot at 18, shade left. Interpretation: 18 or fewer visits. (correct answer)
2. Solve: $5v+10\le100\Rightarrow v\le90$. Graph: closed dot at 90, shade left. Interpretation: 90 or fewer visits.
3. Solve: $5v+10\le100\Rightarrow 5v\le110\Rightarrow v\le22$. Graph: closed dot at 22, shade left. Interpretation: 22 or fewer visits.
4. Solve: $5v+10\le100\Rightarrow 5v\le90\Rightarrow v\ge18$. 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 B is wrong because it skips dividing by 5 after subtracting. Choice C is wrong because it adds 10 instead of subtracting it. Choice D is wrong because it flips the inequality sign, which should only happen when dividing by a negative number.

Question 19

1. Inequality: $15-x\ge25$. Solution: $x\le-10$. Graph: closed dot at -10, shade left.
2. Inequality: $x-15\le25$. Solution: $x\le40$. Graph: closed dot at 40, shade left.
3. Inequality: $x-15\ge25$. Solution: $x\ge40$. Graph: closed dot at 40, shade right. (correct answer)
4. Inequality: $x+15\ge25$. Solution: $x\ge10$. 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 20

1. $t\le2.25$. Graph: closed dot at 2.25, shade left. Interpretation: at most 2 tickets (whole tickets). (correct answer)
2. $t\le2.75$. Graph: closed dot at 2.75, shade left. Interpretation: at most 2 tickets (whole tickets).
3. $t<2.25$. Graph: open circle at 2.25, shade left. Interpretation: at most 2 tickets (whole tickets).
4. $t\le2.25$. 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.