Solve and Graph Linear Inequalities - 7th Grade Math

A class is collecting cans for a food drive. They already collected 18 cans. Each student who helps brings in 6 more cans. The class wants to collect at least 60 cans total. Write, solve, and graph an inequality for the number of helping students s needed. (Students must be whole numbers.)
Answer: 18+6s≥60 ⇒ 6s≥42 ⇒ s≥7; graph: closed dot at 7 and shade right The situation means the class already has 18 cans and needs 6 more cans per helping student to reach at least 60 total, so the inequality is 18 + 6s is greater than or equal to 60. Subtracting 18 from both sides gives 6s is greater than or equal to 42, and dividing both sides by 6, a positive number, keeps the inequality direction the same, giving s is greater than or equal to 7. Since the inequality includes at least, the graph uses a closed dot at 7 with shading to the right, showing 7 or more students. Choice B forgets to divide by 6 after subtracting, leaving an incorrect solution. Choice C changes at least to a strict inequality, using an open dot instead of a closed one, and Choice D flips the inequality direction without a reason to do so.
A school fundraiser sells bracelets. The club already has $-3$ dollars in its cash box (it owes $3) and earns $4 per bracelet sold. How many bracelets must be sold for the cash box amount to be more than $25$ ? Solve and graph: $-3 + 4b > 25$
Answer: $b>7$ . Graph: open circle at 7, shade right. Smallest whole number is 8 bracelets. 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 $px + q \leq r$ : subtract q ( $px \leq r - q$ ), divide by p ( $x \leq (r - q)/p$ if p>0, flip to $x \geq \dots$ if p<0); graphing uses closed dot ● for $\leq$ or $\geq$ (includes boundary), open ○ for $<$ or $>$ (excludes), shade left for $<$ or $\leq$ (toward smaller), right for $>$ or $\geq$ (toward larger); example: $5v + 10 \leq 100$ → $5v \leq 90$ → $v \leq 18$ graphs as closed dot at 18, shaded left for 18 or less. For this fundraiser starting at $-3$ dollars plus $4$ per bracelet to more than $25$ , set up $-3 + 4b > 25$ , solve by adding 3 ( $4b > 28$ ), dividing by 4 ( $b > 7$ ), graph open circle at 7 shaded right, and interpret smallest whole as 8 bracelets. The correct choice A shows $b>7$ , open circle at 7 shaded right, and smallest 8. Errors include closed dot (choice B), wrong direction (choice C with $b<7$ ), or miscalculation (choice D with $b>5.5$ ). Steps include: (1) translate 'more than' to $>$ , (2) solve adding and dividing no flip, (3) graph open right for $>$ , (4) ceiling to 8 for whole numbers. Common mistakes are boundary type or not rounding up for strict inequality.
What does the solution $v \le 18$ mean in this context: A gym charges $\$10$ per month plus $\$5$ per visit, and you want to spend at most $\$100$ in a month. ( $v$ is the number of visits.)
Answer: You can visit the gym 18 or fewer times and still stay within the budget. Solving 10 plus 5v is less than or equal to 100 gives 5v is less than or equal to 90, so v is less than or equal to 18, meaning you can visit the gym 18 or fewer times and still stay within the $100 budget, which matches choice C. Choice B incorrectly excludes the boundary value, treating the solution as if it meant strictly fewer than 18. Choice A reverses the inequality direction, suggesting you must visit at least 18 times. Choice D incorrectly treats the solution as a single fixed value instead of a range of possible visit counts. Since the inequality includes equal to, visiting exactly 18 times should still count as staying within budget.
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: $6m + 4 < 40$
Answer: $m<6$ . Graph: open circle at 6, shade left. Interpretation: fewer than 6 months. 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.
A movie theater offers a membership where you pay $25 upfront and then $8 per movie. Without membership, each movie costs $13. After how many movies will the membership cost less than paying per movie?
Answer: After $6$ movies, membership becomes cheaper than paying per movie Membership cost for $n$ movies: $25 + 8n$ . Regular cost: $13n$ . We want $25 + 8n < 13n$ , which gives $25 < 5n$ , so $n > 5$ . This means after $6$ movies, membership becomes cheaper. Let's verify: At $5$ movies, membership costs $65, regular costs $65 (equal). At $6$ movies, membership costs $73, regular costs $78 (membership cheaper). Choices A, C, and D give incorrect breakeven points.
Verify whether $x=-2$ is a solution to the inequality $3x-5<4$ .
Answer: Yes, because $3(-2)-5=-11$ and $-11<4$ . Substituting x equals negative 2 into 3x minus 5 gives 3 times negative 2 minus 5, which equals negative 6 minus 5, or negative 11, and since negative 11 is less than 4, the inequality is true, so x equals negative 2 is a solution, matching choice D. Choice A contains an arithmetic error, computing the left side as 11 instead of negative 11, which leads to the wrong conclusion. Choice B also miscalculates the left side as 1 instead of negative 11. Choice C correctly computes negative 11 but then wrongly compares it as greater than 4 instead of less than 4. Careful sign handling when multiplying and subtracting negative numbers is the key skill being tested here.
A gym charges a $10 monthly base fee plus $5 per visit. You want to spend at most $100 this month. If $v$ represents the number of visits, which of the following correctly solves the inequality $5v + 10 \le 100$ and correctly describes the graph of the solution?
Answer: Solve: $5v+10\le100\Rightarrow 5v\le90\Rightarrow v\le18$ . Graph: closed dot at 18, shade left. Interpretation: 18 or fewer visits. 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.
A student's final grade is calculated as $0.7T + 0.3F$ where $T$ is the test average and $F$ is the final exam score. If a student has a test average of $78$ and needs a final grade of at least $82$ , what minimum score is needed on the final exam?
Answer: The student needs at least $92$ points on the final exam 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.
A student is saving for headphones. They already have $18 and earn $6 each week doing chores. They want at least $60 total. Let w be the number of weeks. Solve the inequality and graph the solution on a number line.
Answer: 18+6w>=60 => 6w>=42 => w>=7; graph: closed dot at 7, shade right Subtracting 18 from both sides of 18 + 6w >= 60 gives 6w >= 42, and dividing by 6 gives w >= 7, graphed as a closed dot at 7 with shading to the right, matching choice D. Choice A flips the inequality direction incorrectly, landing on w <= 7. Choice B skips the division step, leaving w >= 42. Choice C uses a strict inequality (>) with an open dot instead of the correct >= and closed dot, since 'at least' includes the boundary value.
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 $p$ you can add.
Answer: $0.5p+1.5<5 \Rightarrow 0.5p<3.5 \Rightarrow p<7$ ; graph: open dot at 7 and shade left 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.
A temperature starts at -5 degrees Celsius and rises 3 degrees Celsius per hour. After how many hours will the temperature be greater than 10 degrees Celsius? Solve and graph the solution set on a number line. Inequality: -5 + 3h > 10
Answer: h>5. Graph: open circle at 5, shade right. Interpretation: after more than 5 hours. The temperature starts at -5 degrees and rises 3 degrees each hour, so the inequality is -5 + 3h is greater than 10. Adding 5 to both sides gives 3h is greater than 15, and dividing both sides by 3 gives h is greater than 5, matching Choice B. Since the inequality is strict, the graph uses an open circle at 5 shaded to the right, meaning the temperature passes 10 degrees sometime after the 5-hour mark. Choice A makes an arithmetic error, dividing 15 by 3 incorrectly to get 5/3. Choice C solves the inequality with the wrong direction, giving h less than 5 instead of greater than 5. Choice D uses a closed dot instead of an open one, which would mean exactly 5 hours already reaches more than 10 degrees, but at exactly 5 hours the temperature is exactly 10 degrees, not more.
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 $q$ be the number of quizzes. Solve the inequality and state the smallest whole number of quizzes you must complete. Inequality: $68 + 2q > 75$
Answer: $q>3.5$ , so you need at least $4$ quizzes. 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.
A school club orders custom stickers. There is a setup fee of $15 plus $0.50 per sticker. The club has a maximum budget of $40. Let s be the number of stickers. Solve the inequality and choose the correct solution and graph description.
Answer: 0.5s+15<=40 => 0.5s<=25 => s<=50; graph: closed dot at 50, shade left Since the budget is a maximum, the inequality is 0.5s + 15 <= 40. Subtracting 15 gives 0.5s <= 25, and dividing by 0.5 gives s <= 50, graphed as a closed dot at 50 with shading to the left, matching choice B. Choice A uses a strict inequality (<) with an open dot, but 'maximum' means the budget amount itself is allowed, so the boundary should be included. Choice C reverses the inequality direction, and choice D skips the division by 0.5, landing on the wrong boundary value.
Marcus has $\$85$ saved and earns $\$12$ per hour tutoring. He wants to buy a laptop that costs $\$200$ . However, he also needs to keep at least $\$25$ for bus fare. What is the minimum number of hours Marcus must tutor to afford the laptop while keeping money for bus fare?
Answer: $12$ hours Marcus needs $200 for the laptop plus $25 for bus fare, totaling $225. He starts with $85, so he needs 225 minus 85, or $140 more. Setting up the inequality 85 + 12h >= 225 gives 12h >= 140, so h >= 11.67. Since he can't work partial hours, he needs at least 12 hours. Choice B, 10 hours, gives only $205 total, which isn't enough for both the laptop and bus fare. Choice C, 15 hours, is more tutoring than necessary. Choice D, 9 hours, gives only $193 total, far short of what's needed.
A phone plan charges $40 per month plus $0.15 per text message over $200$ messages. If Chen's monthly bill cannot exceed $55, what is the maximum number of text messages he can send?
Answer: Chen can send at most $300$ text messages per month 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.
The solution to the inequality $\frac{2x - 6}{3} \leq 4$ is graphed on the number line. Which value is NOT in the solution set?
Answer: $x = 10$ Solving $\frac{2x - 6}{3} \leq 4$ : multiply by $3$ to get $2x - 6 \leq 12$ , so $2x \leq 18$ , thus $x \leq 9$ . The graph shows a closed circle at $9$ with shading to the left. Values $7$ , $8$ , and $9$ are all $\leq 9$ , so they're in the solution set. However, $x = 10 > 9$ , so it's not in the solution set.
The number line represents the solution to the inequality $-2x + 7 > 1$ . Which statement about the solution is correct?
Answer: All values less than $3$ are solutions to the inequality Solving $-2x + 7 > 1$ : $-2x > -6$ , so $x < 3$ (inequality flips when dividing by negative number). The graph shows an open circle at $3$ with shading to the left, confirming $x < 3$ . Choice A incorrectly states $x > 3$ . Choice C is wrong because the open circle means $x = 3$ is not included. Choice D incorrectly includes $x = 3$ and has the wrong direction.
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?
Answer: $8$ miles 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.
A delivery service charges $12 plus $3 per mile for distances over $5$ miles. For distances of $5$ miles or less, they charge a flat $18. Maria has $30 to spend and needs to travel more than $5$ miles. What is the maximum distance she can travel?
Answer: Maria can travel at most $11$ miles with her budget Since Maria travels more than 5 miles, her cost is 12 plus 3 times the distance beyond 5 miles, so the inequality is 12 plus 3 times the quantity d minus 5, which must be less than or equal to 30. Solving gives 3 times the quantity d minus 5 is less than or equal to 18, so d minus 5 is less than or equal to 6, and therefore d is less than or equal to 11, meaning the maximum distance Maria can travel is 11 miles, matching choice D. Checking this answer, at 11 miles the cost is 12 plus 3 times 6, or 12 plus 18, which equals exactly $30, using the full budget. Choices A, B, and C, at 6, 7, and 8 miles, are all well within budget but do not represent the true maximum distance Maria can afford. Solving the full inequality all the way through, rather than stopping at an early step, is the only reliable way to find the actual maximum distance.
A temperature starts at $-5^\circ\text{C}$ and rises $3^\circ\text{C}$ per hour. After how many hours will the temperature be greater than $10^\circ\text{C}$ ? Solve and graph the solution set on a number line. Inequality: $-5 + 3h > 10$
Answer: $h>5$ . Graph: open circle at 5, shade right. Interpretation: after more than 5 hours. 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.

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