Linear Inequalities - PSAT Math

1. A club requires members to be at least 16 years old. Let $a$ be a person's age. Which inequality represents the requirement?
   Answer: $a\ge16$ This question asks for the inequality representing the age requirement of at least 16 for club membership. Let a be age, so a ≥ 16 models 'at least'. No solving needed. A common error is using > instead of ≥, excluding 16. Another mistake is reversing to ≤ or <. Include equality for 'at least' unless specified as more than.
2. Solve the compound inequality $2x+1<9$ AND $x-3\ge-1$. Which solution is correct?
   Answer: $2\le x<4$ The question asks to solve the compound inequality 2x + 1 < 9 AND x - 3 ≥ -1 and identify the correct solution set. Solve each part separately: for 2x + 1 < 9, subtract 1 to get 2x < 8, then divide by 2 to obtain x < 4. For x - 3 ≥ -1, add 3 to get x ≥ 2. The intersection is 2 ≤ x < 4. A common error is confusing AND with OR, leading to a union instead of intersection. Another pitfall is not handling the boundaries correctly, such as making both inequalities strict. When dealing with compound inequalities, graph each part on a number line to visualize the overlap.
3. Solve the compound inequality $-1\le 2x+3<7$. Which solution set is correct?
   Answer: $-2\le x<2$ This question involves solving the compound inequality -1 ≤ 2x + 3 < 7 and selecting the correct solution set. Subtract 3 from all parts to get -4 ≤ 2x < 4. Then divide by 2, which is positive so no reversal, resulting in -2 ≤ x < 2. A key error is incorrect subtraction, leading to wrong bounds like in choice C. Another mistake is changing inequality directions unnecessarily, as seen in choice D. For compound inequalities, handle all parts simultaneously and test a value like x=0 to confirm it satisfies the original.
4. A number line graph shows an open circle at $-2$ and shading to the right. Which inequality matches the graph?
   Answer: $x>-2$ The question asks you to match a number line graph with an open circle at -2 and shading to the right to the correct inequality. An open circle indicates the endpoint is not included, and shading to the right means values greater than -2, so the inequality is x > -2. This differs from a closed circle, which would use ≥ or ≤. A common error is confusing the direction of shading, leading to choices like C or D that point left. Another pitfall is misinterpreting the open circle as inclusive, which might suggest choice A instead of B. When interpreting graphs, note that open circles pair with strict inequalities like > or <.
5. Solve the compound inequality $3x-6>0$ AND $x+2\le7$. Which solution is correct?
   Answer: $2<x\le5$ The task is to solve the compound inequality 3x - 6 > 0 AND x + 2 ≤ 7, finding the combined solution. First, solve 3x - 6 > 0 by adding 6 and dividing by 3 (positive, so no reversal), getting x > 2. Next, solve x + 2 ≤ 7 by subtracting 2, resulting in x ≤ 5. The intersection is 2 < x ≤ 5, which is choice C. Errors often occur by treating it as OR instead of AND, leading to unions like in D, or mishandling the inequalities separately. When solving compounds, graph each part on a number line to visualize the overlap. A useful strategy is to check endpoints: x=2 fails the strict inequality, but x=5 works.
6. Solve the compound inequality $2x-7<-1$ AND $x+4\ge 6$. Which set of values satisfies both conditions?
   Answer: $2\le x<3$ This question involves solving the compound inequality 2x - 7 < -1 AND x + 4 ≥ 6. Solve first: 2x < 6, so x < 3. Solve second: x ≥ 2. Intersection is 2 ≤ x < 3. Choice D represents 'or,' not 'and,' and A ignores the second condition. A common pitfall is solving one part and forgetting the intersection for 'and.' Another error is not handling the compound nature, leading to separate solutions. When dealing with 'and,' find overlapping values and test points in both originals.
7. A phone plan costs $25 per month plus $0.10 per text message. If your monthly bill must be less than $40, what is the greatest number of texts $x$ you can send while staying under the limit?
   Answer: $x=149$ We need to find the maximum number of texts when the total bill must be less than $40. The inequality is $25 + 0.10x < 40$. Subtracting 25 from both sides gives $0.10x < 15$, then dividing by 0.10 yields $x < 150$. Since x must be a whole number and the inequality is strict (<), the greatest integer value is 149. A common mistake is using 150, but that would make the bill exactly $40, violating the "less than" requirement. When dealing with strict inequalities and whole numbers, the answer is one less than the boundary value.
8. A number line shows a closed circle at $-3$ and shading to the right. Which inequality matches the graph?
   Answer: $x\ge-3$ This question asks for the inequality matching a number line with a closed circle at -3 and shading to the right. Closed circle includes -3, right shading means greater than or equal, so x ≥ -3. Direct from graph. A common error is using > for closed circle. Another mistake is left shading as less than. Associate closed circles with ≥ or ≤.
9. A gym charges a one-time sign-up fee of $15 plus $8 per visit. You have at most $79 to spend this month. Let $v$ be the number of visits. Which inequality represents the situation?
   Answer: $8v+15\le79$ This question asks for the inequality representing the gym cost situation where you have at most $79 to spend on a $15 fee plus $8 per visit. The total cost is modeled as $15 + 8v$, and since it's at most $79, the inequality is $15 + 8v \le 79$, which matches choice B after rearranging. No multiplication or division by negative occurs here, so the inequality direction remains the same. A common error is using $\ge$ instead of $\le$, as in choice A, which would imply at least $79. Another pitfall is swapping the coefficients, like in choice D, or misdistributing as in choice C. When modeling constraints, clearly distinguish between 'at most' meaning $\le$ and 'at least' meaning $\ge$.
10. Solve the inequality $\dfrac{x}{4}-3\ge2$. What is the solution?
    Answer: $x\ge20$ The question requires solving the inequality (x/4) - 3 ≥ 2 and identifying the solution set. Start by adding 3 to both sides to isolate the term with x, yielding x/4 ≥ 5. Then, multiply both sides by 4, which is positive, so the inequality direction remains the same, resulting in x ≥ 20. This corresponds to choice A. A key error to avoid is forgetting to adjust the inequality when multiplying or dividing by a negative number, but here it's positive, so no reversal is needed. Students might mistakenly subtract instead of adding, leading to incorrect options like C or D. For inequalities, always verify your solution by plugging in a test value, such as x=20, which satisfies the original inequality.
11. Solve the inequality $-3(2x-5)\ge 9$. Which of the following gives the correct solution set for $x$?
    Answer: $x\le 1$ This question requires solving the inequality -3(2x - 5) ≥ 9 and identifying the correct solution set for x. Start by distributing the -3 to get -6x + 15 ≥ 9, then subtract 15 from both sides to obtain -6x ≥ -6. Now divide both sides by -6, and since you're dividing by a negative number, you must reverse the inequality symbol, resulting in x ≤ 1. A key error is forgetting to reverse the inequality when dividing by a negative, which might lead to x ≥ 1 as in choice B. Another common mistake is mishandling the distribution, potentially leading to incorrect bounds like in choices C or D. For inequalities involving negatives, always verify by testing a value in the solution set, such as x=0 which satisfies the original inequality.
12. A fundraiser sells candles for $9 each. The group must raise more than $270. Let $c$ be the number of candles sold. Which inequality represents the goal?
    Answer: $9c>270$ This question asks for the inequality representing selling candles at $9 each to raise more than $270. Total is 9c, and more than means $9c > 270$. No negatives here. A key error is using $\ge$ instead of $>$, including exactly 270. Another pitfall is omitting the 9, like $c > 270$. Use strict inequality for 'more than'.
13. A theater must sell at least 120 tickets to break even. It has already sold 45 tickets. Let $t$ be the additional tickets sold. Which inequality shows how many more tickets must be sold?
    Answer: $t+45\ge120$ This question asks for the inequality showing how many additional tickets t must be sold to reach at least 120 total, with 45 already sold. The total tickets are 45 + t, and it must be at least 120, so 45 + t ≥ 120. No negative multiplication occurs, so the direction stays the same. A common error is using ≤ instead of ≥, as if it were at most. Another mistake is subtracting 45 incorrectly, like in choice B or D. Carefully match words like 'at least' to ≥ in modeling problems.
14. A delivery truck can carry at most 1,200 pounds. The truck already has 350 pounds loaded, and each box weighs 55 pounds. Let $x$ be the number of boxes added. Which inequality shows the possible values of $x$?
    Answer: $55x+350\le1200$ This question asks for the inequality modeling added boxes without exceeding 1,200 pounds, with 350 already loaded. Total weight is 55x + 350, and 'at most' means ≤1,200. So, 55x + 350 ≤ 1,200. Choice A uses ≥, opposite of capacity limit, and C subtracts 350 incorrectly. A common error is subtracting instead of adding the initial load. Another pitfall is using < instead of ≤, excluding the maximum. When modeling limits, verify by considering if equality is allowed, like exactly 1,200 pounds.
15. Solve the inequality $\dfrac{-3x+12}{3}>1$. What is the solution?
    Answer: $x<3$ The question asks to solve (-3x + 12)/3 > 1. Multiply by 3 to -3x + 12 > 3, subtract 12 to -3x > -9, divide by -3 and flip to x < 3. Flipping is key with negative division. A common error is not flipping, leading to x > 3. Another is order of operations. Always note negative coefficients.
16. A storage bin can hold no more than 60 pounds. A box weighs 12 pounds and each book weighs 3 pounds. Let $b$ be the number of books. Which inequality models the constraint?
    Answer: $12+3b\le60$ This question asks for the inequality modeling the weight constraint of a bin holding no more than 60 pounds, with a 12-pound box and 3-pound books. The total weight is 12 + 3b, and it must be ≤ 60. No negative operations are involved here. A common error is using ≥ instead of ≤, as if it were at least 60. Another mistake is incorrect modeling, like subtracting or multiplying wrongly as in choices B or D. Verify the inequality direction matches phrases like 'no more than' to ≤.
17. A runner wants to run more than 20 miles this week. She has already run 7.2 miles and 6.5 miles. Let $x$ be miles she must run in the remaining days. Which inequality represents the requirement?
    Answer: $x>6.3$ The question requires an inequality for miles x to run more than 20 total, with 7.2 and 6.5 already. Total 13.7 + x > 20, so x > 6.3. Uses > for 'more than.' A key error is using ≥ for strict more. Another is addition error. Use strict inequalities for 'more than.'
18. A number line shows an open circle at $0$ and shading to the left. Which inequality matches the graph?
    Answer: $x<0$ This question asks for the inequality matching a number line with an open circle at 0 and shading to the left. Open circle excludes 0, left shading means less than, so x < 0. Direct interpretation. A key error is using ≤ for open circle. Another pitfall is right shading as greater than. Open circles mean strict < or >.
19. A number line shows an open circle at $-4$ and shading to the right. Which inequality matches the graph?
    Answer: $x > -4$ This question asks which inequality matches a number line with an open circle at $-4$ and shading to the right. An open circle indicates the boundary is not included, and shading right means values greater than $-4$. Thus, the inequality is $x > -4$, corresponding to choice B. A common pitfall is confusing open with closed circles, leading to $\geq$ or $\leq$ like in A or C, or misreading the direction of shading. Remember, left shading means less than, right means greater than. When no operations involve negatives, inequality direction isn't an issue here. For graphs, always note if the circle is open (strict) or closed (inclusive) to choose the correct symbol.
20. A runner wants to average at least 8 miles per hour. If the runner runs 6 miles in $t$ hours, which inequality represents the requirement?
    Answer: $\dfrac{6}{t}\ge 8$ To find the inequality for averaging at least 8 miles per hour, we use the formula: speed = distance/time. The runner's speed is 6/t miles per hour, and this must be at least 8, giving us $6/t ≥ 8$. Note that we don't multiply or divide by t here because we don't know if t is positive (though logically it must be for time). The common error is inverting the fraction or the inequality direction. When setting up rate problems, carefully identify what goes in the numerator and denominator.

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