PSAT Math Quiz: Linear Inequalities

What this quiz covers

This quiz focuses on Linear Inequalities, giving you a quick way to practice the rules, question types, and explanations that matter most for PSAT 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. $x\le0$
2. $x<0$ (correct answer)
3. $x\ge0$
4. $x>0$

Explanation: 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 >.

Question 2

1. $6x+27\ge 80$ (correct answer)
2. $6x+27\le 80$
3. $6x-27\ge 80$
4. $27-6x\ge 80$

Explanation: This question requires an inequality for selling at least 80 items total, with 27 already sold and 6 per day for x days. Total items: 6x + 27, and 'at least' means ≥80. So, 6x + 27 ≥ 80. Choice B uses ≤, which is the opposite, and D rearranges terms incorrectly. A key error is using subtraction instead of addition for already sold items. Another pitfall is confusing 'at least' with 'at most.' For goal-oriented inequalities, ensure the symbol matches 'at least' (≥) and test with x=0 to check baseline.

Question 3

1. $t+45\le120$
2. $t-45\ge120$
3. $t+45\ge120$ (correct answer)
4. $45-t\ge120$

Explanation: 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.

Question 4

1. $s\ge34$
2. $s\le34$ (correct answer)
3. $18s\le52$
4. $52-s\le18$

Explanation: The question requires an inequality for additional students s on a bus with at most 52 total and 18 already aboard. The situation is 18 + s ≤ 52, which simplifies to s ≤ 34 by subtracting 18. This uses ≤ because 'at most' includes equality. A key error is using greater than for capacity limits. Another mistake is adding instead of subtracting the existing students. In modeling problems, define variables clearly and choose the correct inequality direction based on limits.

Question 5

1. $x>-3$
2. $x\ge-3$ (correct answer)
3. $x<-3$
4. $x\le-3$

Explanation: 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 ≤.

Question 6

1. $x<8$ (correct answer)
2. $x\le8$
3. $x>8$
4. $x\ge8$

Explanation: The task is to solve $(2x - 1)/3 < 5$ and find the solution. Multiply both sides by 3 (positive, no reversal) to get $2x - 1 < 15$. Add 1: $2x < 16$, then divide by 2 (positive, no reversal): $x < 8$. This is choice A. Common mistakes include using $\le$ for non-strict inequalities, as in B, or reversing due to forgotten steps. If there were negatives, explicitly reverse, but here all operations preserve direction. Students might subtract instead of adding, leading to $>$ like in C. A strategy is to work backwards from choices by testing values like $x=7$ in the original inequality.

Question 7

1. $x\le21$ (correct answer)
2. $x\ge21$
3. $x\le-21$
4. $x\ge-21$

Explanation: The question asks to solve 9 - (x/3) ≥ 2 and find the values of x that make it true. Add (x/3) to both sides to get 9 ≥ 2 + (x/3), then subtract 2 to obtain 7 ≥ x/3. Multiply both sides by 3, which is positive, so the inequality remains ≥, resulting in 21 ≥ x or x ≤ 21. A common error is flipping the inequality when multiplying by positive 3. Another pitfall is subtracting incorrectly from both sides. To verify, test boundary values like x = 21 in the original inequality.

Question 8

1. $x\le 6$ (correct answer)
2. $x<6$
3. $x\ge 6$
4. $x\le 14$

Explanation: This problem asks us to find the maximum number of exhibits you can visit with a $30 budget when there's a $12 entry fee and $3 per exhibit. We need to set up an inequality: total cost ≤ $30, which gives us $12 + 3x ≤ 30$. Solving this, we subtract 12 from both sides to get $3x ≤ 18$, then divide by 3 to get $x ≤ 6$. The key error to avoid is forgetting to include the entry fee or using a strict inequality when the problem says "at most." When dealing with real-world constraints involving money, remember that "at most" translates to ≤, not <.

Question 9

1. $120\le x\le 160$
2. $120<x\le 160$ (correct answer)
3. $120<x<160$
4. $x\le 120\ \text{or}\ x>160$

Explanation: This problem describes production constraints where output must be more than 120 parts per hour but at most 160 parts per hour. "More than 120" translates to $x > 120$ (strict inequality), while "at most 160" means $x ≤ 160$ (inclusive). Combining these gives us $120 < x ≤ 160$. The critical distinction is recognizing when to use strict versus inclusive inequalities based on the wording. Pay attention to phrases like "more than" (>) versus "at least" (≥) to avoid boundary errors.

Question 10

1. $x\le-3$
2. $x\ge-3$ (correct answer)
3. $x\le3$
4. $x\ge3$

Explanation: This question asks for the values of x satisfying (3x + 1)/(-2) ≤ 4. Multiply both sides by -2, and since negative, reverse ≤ to ≥, getting 3x + 1 ≥ -8. Subtract 1 to get 3x ≥ -9, divide by 3 (positive, no reversal) to get x ≥ -3. A common error is not reversing when multiplying by negative, leading to x ≤ -3. Another mistake is arithmetic, getting x ≥ 3. Test boundary values to confirm.

Question 11

1. $55x+350\ge1200$
2. $55x+350\le1200$ (correct answer)
3. $55x-350\le1200$
4. $55x+350<1200$

Explanation: 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.

Question 12

1. $b\ge 8$
2. $8\le b\le 16$ (correct answer)
3. $8<b<16$
4. $b\le 8\text{ or }b\ge 16$

Explanation: This question involves creating a compound inequality for the number of granola bars b, where at least 8 are needed and the cost of $1.50 each is at most $24. The minimum gives b ≥ 8, and for the cost, 1.5b ≤ 24, so divide by 1.5 to get b ≤ 16, combining to 8 ≤ b ≤ 16. Since b is likely whole numbers, possible values are integers from 8 to 16. A key error is using strict inequalities like in choice C, which excludes the endpoints that are valid here. Another mistake is inverting the inequalities, leading to outside ranges like in choice D. For compound inequalities, test boundary values to ensure they satisfy both conditions.

Question 13

1. $x\le 10$
2. $x\ge 10$
3. $x\le 20$ (correct answer)
4. $x\ge 20$

Explanation: We need to solve $(x-5)/3 + 2 ≤ 7$ for x. First, subtract 2 from both sides to get $(x-5)/3 ≤ 5$. Multiply both sides by 3 to get $x - 5 ≤ 15$. Finally, add 5 to both sides to obtain $x ≤ 20$. The key is to perform operations systematically, treating the inequality like an equation but being careful about sign changes. When fractions are involved, clear them early by multiplying to simplify your work.

Question 14

1. $x\le3$
2. $x<3$ (correct answer)
3. $x\ge3$
4. $x>3$

Explanation: This question requires matching an inequality to a number line with an open circle at 3 and shading to the left. The open circle means 3 is excluded, and left shading indicates values less than 3. Therefore, the inequality is x < 3, which is choice B. Errors often involve mistaking open for closed, leading to ≤ or ≥ like in A or C, or misinterpreting direction. No negative operations here, so inequality direction isn't flipped. A pitfall is confusing left with greater than. When interpreting graphs, remember open means strict and direction dictates less/greater; verify by picking a shaded point like x=2.

Question 15

1. $9c\ge270$
2. $9c>270$ (correct answer)
3. $c>270$
4. $c\ge30$

Explanation: 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'.

Question 16

1. $12x\ge 240$
2. $12x\le 240$
3. $6\le x\le 20$ (correct answer)
4. $6\le x<20$

Explanation: The question asks for a compound inequality for the number of 12-seat blocks x in a 240-seat theater, with at least 6 blocks and no more than the total seats allow. This gives x ≥ 6 and 12x ≤ 240, dividing by 12 for x ≤ 20, so 6 ≤ x ≤ 20. Assuming x is an integer, values are from 6 to 20. A common error is using single inequalities like in A or B instead of compound. Another pitfall is strict inequality like in D, excluding valid endpoints. When building inequalities from constraints, combine them and check endpoints satisfy the conditions.

Question 17

1. $-6$
2. $-2$
3. $6$
4. $10$ (correct answer)

Explanation: This question requires finding which given value of x satisfies the inequality 5 - 2x ≤ -9. Solve by subtracting 5 to get -2x ≤ -14, then divide by -2, reversing the inequality since dividing by negative, yielding x ≥ 7. Among the choices, only x=10 satisfies, as 5 - 2(10) = -15 ≤ -9 is true, while others like x=6 give -7 > -9. A key error is not reversing the inequality, which would incorrectly suggest x ≤ 7 and pick choice C. Another mistake is plugging in without solving, potentially missing the correct one. For verification questions, substitute each choice into the original inequality to confirm.

Question 18

1. $12v\ge85$ (correct answer)
2. $12v\le85$
3. $v+12\ge85$
4. $v+12\le85$

Explanation: When you encounter word problems involving inequalities, focus on identifying the constraint and translating it into mathematical language. Here, you need to determine how many vans are required to seat at least 85 people. Let's set up the relationship: each van seats 12 people, so $v$ vans can seat $12v$ people total. Since at least 85 members are going, the total seating capacity must be greater than or equal to 85. This gives us $12v \geq 85$. Choice A ($12v \geq 85$) correctly represents this situation. The left side shows total seating capacity, and the inequality ensures this capacity meets or exceeds the 85-member requirement. Choice B ($12v \leq 85$) reverses the inequality direction, suggesting the seating capacity should be less than or equal to 85 people. This makes no sense since you need to accommodate at least 85 members. Choice C ($v + 12 \geq 85$) incorrectly adds the number of vans to the seating capacity per van, rather than multiplying. This would mean $v + 12$ people can be seated, which doesn't reflect how van capacity actually works. Choice D ($v + 12 \leq 85$) combines both errors from choices B and C: it uses addition instead of multiplication and reverses the inequality direction. Key strategy: In constraint problems, identify what quantity must meet or exceed a threshold, then ensure your inequality points in the correct direction. The phrase "at least" always translates to "greater than or equal to" (≥).

Question 19

1. $2<f<5$
2. $2\le f\le5$ (correct answer)
3. $f\le2\ \text{or}\ f\ge5$
4. $2\le f<5$

Explanation: This question asks for the inequality constraining flour f between 2 and 5 cups, inclusive. Inclusive means 2 ≤ f ≤ 5. No solving involved. A key error is using strict inequalities, excluding 2 and 5. Another pitfall is 'or' instead of between. Include equalities for 'inclusive'.

Question 20

1. $x> -1$
2. $x< -1$ (correct answer)
3. $x>5$
4. $x<5$

Explanation: The question asks to solve -6(x - 2) > 18. First, distribute -6 to get -6x + 12 > 18, then subtract 12 to obtain -6x > 6. Divide by -6, which is negative, so reverse the inequality to x < -1. The key step is flipping the > to < when dividing by negative. A common error is forgetting to flip the inequality sign. Another pitfall is incorrect distribution of the negative. Always highlight operations with negatives to avoid sign errors in inequalities.