Solving Linear Inequalities - Accuplacer Aaf

1. Which of the following represents the solution to the inequality $3(x - 4) - 5x \geq -2$?
   Answer: $x \leq -5$ To solve the inequality, first distribute the 3: $3x - 12 - 5x \geq -2$. Combine like terms on the left side: $-2x - 12 \geq -2$. Add 12 to both sides: $-2x \geq 10$. Finally, divide both sides by -2. Remember to reverse the inequality sign when dividing by a negative number: $x \leq -5$.
2. Which of the following is the solution to $0.5(x - 3) \geq 0.2x + 1.2$?
   Answer: $x \geq 9$ First, distribute 0.5 on the left side: $0.5x - 1.5 \geq 0.2x + 1.2$. Subtract $0.2x$ from both sides: $0.3x - 1.5 \geq 1.2$. Add 1.5 to both sides: $0.3x \geq 2.7$. Divide both sides by 0.3: $x \geq 9$. Alternatively, one could multiply the entire inequality by 10 at the start to eliminate the decimals.
3. The solution set for a linear inequality is given by $x \leq -4$. Which of the following inequalities has this solution set?
   Answer: $5 - x \geq 9$ Solve each inequality. For A: $8 \leq 2x$, so $x \geq 4$. For B: $-x \geq 4$. Multiplying by -1 and reversing the inequality sign gives $x \leq -4$. This matches the given solution set. For C: $-3x \leq 12$, so $x \geq -4$. For D: $12 \leq 3x$, so $x \geq 4$.
4. What is the solution set for the compound inequality $3x - 5 \leq 4(x - 1) - x < 2x + 7$?
   Answer: $x < 11$ First, simplify the middle expression: $4(x-1) - x = 4x - 4 - x = 3x - 4$. The inequality becomes $3x - 5 \leq 3x - 4 < 2x + 7$. This must be broken into two parts. Part 1: $3x - 5 \leq 3x - 4$. Subtracting $3x$ gives $-5 \leq -4$, which is always true. Part 2: $3x - 4 < 2x + 7$. Subtracting $2x$ gives $x - 4 < 7$. Adding 4 gives $x < 11$. The overall solution is the intersection of the solutions to both parts. Since Part 1 is true for all real numbers, the solution is determined entirely by Part 2, which is $x < 11$.
5. Which of the following is the solution to the inequality $\frac{x - 2}{3} - \frac{x - 1}{4} \leq 1$?
   Answer: $x \leq 17$ To clear the denominators, multiply the entire inequality by the least common denominator of 3 and 4, which is 12: $12\left(\frac{x - 2}{3}\right) - 12\left(\frac{x - 1}{4}\right) \leq 12(1)$. This simplifies to $4(x - 2) - 3(x - 1) \leq 12$. Distribute the 4 and -3: $4x - 8 - 3x + 3 \leq 12$. Combine like terms on the left: $x - 5 \leq 12$. Add 5 to both sides: $x \leq 17$.
6. What is the solution set for the inequality $5x - 4(x + 1) \leq x - 5$?
   Answer: No solution First, distribute the -4 on the left side: $5x - 4x - 4 \leq x - 5$. Combine like terms on the left side: $x - 4 \leq x - 5$. Subtract $x$ from both sides: $-4 \leq -5$. This statement is always false. Since there is no value of $x$ that can make this false statement true, there is no solution.
7. What is the solution to the inequality $\frac{2}{3}x - \frac{1}{2} > \frac{1}{6}x + 2$?
   Answer: $x > 5$ To eliminate the fractions, multiply the entire inequality by the least common denominator, which is 6. This gives: $6(\frac{2}{3}x) - 6(\frac{1}{2}) > 6(\frac{1}{6}x) + 6(2)$, which simplifies to $4x - 3 > x + 12$. Subtract $x$ from both sides: $3x - 3 > 12$. Add 3 to both sides: $3x > 15$. Divide by 3: $x > 5$.
8. What is the smallest integer value of $n$ such that $5(n - 2) + 3 > 2n + 14$?
   Answer: 8 First, solve the inequality for $n$. Distribute the 5: $5n - 10 + 3 > 2n + 14$. Combine constants on the left: $5n - 7 > 2n + 14$. Subtract $2n$ from both sides: $3n - 7 > 14$. Add 7 to both sides: $3n > 21$. Divide by 3: $n > 7$. The question asks for the smallest integer value of $n$ that satisfies this condition. The smallest integer greater than 7 is 8.
9. A rental car company charges a daily fee of $25 plus $0.15 per mile driven. If Maria's budget allows her to spend at most $58 per day, what is the maximum number of miles she can drive?
   Answer: 220 miles This is a linear inequality problem where you need to set up an equation based on the cost structure and solve for the maximum value that stays within the budget constraint. The rental company's pricing has two components: a fixed daily fee of $25 plus a variable cost of $0.15 per mile. If Maria drives $x$ miles, her total cost will be $25 + 0.15x$. Since her budget is at most $58, you need to solve: $25 + 0.15x \leq 58$. Subtracting 25 from both sides: $0.15x \leq 33$. Dividing by 0.15: $x \leq 220$. Therefore, Maria can drive a maximum of 220 miles. Let's verify: $25 + 0.15(220) = 25 + 33 = 58$, which exactly matches her budget. Looking at the wrong answers: Choice (A) 225 miles would cost $25 + 0.15(225) = 25 + 33.75 = 58.75$, exceeding her $58 budget. Choice (C) 233 miles would cost $25 + 0.15(233) = 25 + 34.95 = 59.95$, which is nearly $2 over budget. Choice (D) 387 miles would cost $25 + 0.15(387) = 25 + 58.05 = 83.05$, dramatically exceeding her budget. This choice likely comes from incorrectly dividing $58 by $0.15 while ignoring the $25 fixed fee. Strategy tip: In cost problems with fixed and variable components, always set up the inequality as "fixed cost + (rate × variable) ≤ budget limit" and solve step by step. Double-check by substituting your answer back into the original cost formula.
10. A student's scores on three exams are 85, 92, and 88. What is the minimum score the student must earn on a fourth exam to achieve an average of at least 90?
    Answer: 95 Let $x$ be the score on the fourth exam. The average of the four scores is their sum divided by 4. To have an average of at least 90, the inequality is $\frac{85 + 92 + 88 + x}{4} \geq 90$. Simplify the sum: $\frac{265 + x}{4} \geq 90$. Multiply both sides by 4: $265 + x \geq 360$. Subtract 265 from both sides: $x \geq 95$. The minimum score required is 95.
11. If $-2(3x + 4) > 5x - 1$, which of the following represents the solution set?
    Answer: $x < -\frac{7}{11}$ When you encounter an inequality with variables on both sides, you need to isolate the variable while carefully tracking the inequality direction. The key rule to remember is that multiplying or dividing by a negative number flips the inequality sign. Let's solve $-2(3x + 4) > 5x - 1$ step by step. First, distribute the -2: $-6x - 8 > 5x - 1$. Next, collect all x terms on one side by subtracting 5x from both sides: $-6x - 5x - 8 > -1$, which gives us $-11x - 8 > -1$. Add 8 to both sides: $-11x > 7$. Finally, divide both sides by -11, and since we're dividing by a negative number, we must flip the inequality sign: $x < -\frac{7}{11}$. Looking at the wrong answers: Choice A gives $x < -\frac{9}{11}$, which likely results from an arithmetic error when combining terms or working with the fractions. Choice B shows $x > -\frac{7}{11}$, which happens if you forget to flip the inequality sign when dividing by -11. Choice D gives $x > -\frac{9}{11}$, combining both the sign-flipping error and an arithmetic mistake. The correct answer is C: $x < -\frac{7}{11}$. Study tip: Always double-check your work by substituting a test value back into the original inequality. Pick a number that satisfies your solution and verify it works—this catches both arithmetic errors and sign-flipping mistakes.
12. What is the solution set for the absolute value inequality $|2x - 5| \leq 3$?
    Answer: $1 \leq x \leq 4$ The inequality $|A| \leq B$ is equivalent to the compound inequality $-B \leq A \leq B$. Applying this, we get $-3 \leq 2x - 5 \leq 3$. To solve for $x$, first add 5 to all three parts: $-3 + 5 \leq 2x - 5 + 5 \leq 3 + 5$, which simplifies to $2 \leq 2x \leq 8$. Then, divide all three parts by 2: $1 \leq x \leq 4$.
13. Which of the following is the solution set for the compound inequality $-5 \leq 2x + 1 < 7$?
    Answer: $-3 \leq x < 3$ To solve the compound inequality, perform the same operation on all three parts. First, subtract 1 from all parts: $-5 - 1 \leq 2x + 1 - 1 < 7 - 1$, which simplifies to $-6 \leq 2x < 6$. Next, divide all parts by 2: $\frac{-6}{2} \leq \frac{2x}{2} < \frac{6}{2}$, which gives the solution $-3 \leq x < 3$.
14. A salesperson earns a base salary of $1,200 per month plus a 5% commission on sales over $10,000. What is the minimum amount of sales, $S$, the salesperson must make in a month to have a total income of at least $3,000?
    Answer: $46,000 Let $S$ be the total sales. The commission is paid on the amount of sales exceeding $10,000, which is $S - 10,000$. The total income is the base salary plus the commission: $1200 + 0.05(S - 10000)$. This income must be at least $3,000, so we set up the inequality: $1200 + 0.05(S - 10000) \geq 3000$. Subtract 1200 from both sides: $0.05(S - 10000) \geq 1800$. Divide by 0.05: $S - 10000 \geq 36000$. Add 10000 to both sides: $S \geq 46000$. The minimum sales are $46,000.
15. A company's daily cost to produce $x$ items is $C = 30x + 600$. The company sells each item for $70. What is the minimum number of items the company must sell in a day to make a profit?
    Answer: 16 Profit is Revenue minus Cost. The revenue is $R = 70x$. To make a profit, Revenue must be greater than Cost: $R > C$. So, $70x > 30x + 600$. Subtract $30x$ from both sides: $40x > 600$. Divide by 40: $x > 15$. Since the number of items must be an integer and greater than 15, the minimum number of items is 16.
16. Which of the following represents the solution to $|3x + 2| > 8$?
    Answer: $x < -\frac{10}{3}$ or $x > 2$ The inequality $|A| > B$ is equivalent to the compound inequality $A > B$ or $A < -B$. Applying this, we get two separate inequalities: $3x + 2 > 8$ or $3x + 2 < -8$. Solving the first one: $3x > 6$, which gives $x > 2$. Solving the second one: $3x < -10$, which gives $x < -\frac{10}{3}$. The solution is the union of these two sets: $x < -\frac{10}{3}$ or $x > 2$.
17. What is the solution set for the inequality $3|x - 2| - 4 > 5$?
    Answer: $x < -1$ or $x > 5$ First, isolate the absolute value expression. Add 4 to both sides: $3|x - 2| > 9$. Divide by 3: $|x - 2| > 3$. This inequality holds if $x - 2 > 3$ or if $x - 2 < -3$. For the first case, add 2 to both sides to get $x > 5$. For the second case, add 2 to both sides to get $x < -1$. The solution is the union of these two results: $x < -1$ or $x > 5$.
18. A student solved the inequality $-5x + 2 \geq 17$ and got $x \geq -3$. What error did the student most likely make?
    Answer: Failed to reverse the inequality sign when dividing by a negative number When solving inequalities involving negative coefficients, you must carefully track when to reverse the inequality sign. Let's work through this problem step by step to identify the error. Starting with $-5x + 2 \geq 17$, the correct solution process is: - Subtract 2 from both sides: $-5x \geq 15$ - Divide both sides by $-5$: Since you're dividing by a negative number, you must reverse the inequality sign - This gives: $x \leq -3$ The student got $x \geq -3$, which means they performed all the arithmetic correctly but failed to flip the inequality sign when dividing by the negative coefficient. Looking at the answer choices: Choice A is incorrect because the student would have gotten $x \leq -5$ or similar if they miscalculated $17 - 2$. Choice B doesn't make sense because adding $5x$ to both sides first would create $2 \geq 17 + 5x$, leading to a completely different solution path. Choice D is wrong because the negative sign is already correctly distributed in the original expression $-5x + 2$. Choice C correctly identifies the error: the student forgot the fundamental rule that multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign. Study tip: Always double-check your final answer by testing a value. Here, try $x = 0$: substituting into the original inequality gives $2 \geq 17$, which is false, confirming that $x \leq -3$ is correct, not $x \geq -3$.
19. A company's profit $P$ (in thousands of dollars) is modeled by $P = 3x - 15$, where $x$ is the number of units sold (in hundreds). For what values of $x$ will the company have a profit of at least $12,000?
    Answer: $x \geq 9$ Since profit must be at least $12,000 and P is in thousands, we need P ≥ 12. Setting up the inequality: 3x - 15 ≥ 12. Adding 15 to both sides: 3x ≥ 27. Dividing by 3: x ≥ 9. Choice B incorrectly uses 15 from the constant term. Choice C uses 27 without dividing by 3. Choice D converts $12,000 to hundreds instead of recognizing P is already in thousands.
20. What is the solution set for the inequality $4x - 1 < -9$ or $2x + 5 \geq 7$?
    Answer: $x < -2$ or $x \geq 1$ Solve each inequality separately. For the first inequality, $4x - 1 < -9$, add 1 to both sides to get $4x < -8$, then divide by 4 to get $x < -2$. For the second inequality, $2x + 5 \geq 7$, subtract 5 from both sides to get $2x \geq 2$, then divide by 2 to get $x \geq 1$. Since the inequalities are joined by 'or', the solution set includes all numbers that satisfy either condition: $x < -2$ or $x \geq 1$.

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