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📝 Solve inequalities using the Multiplication Property of Inequality (15 MCQs)

📖 From Digital SAT Algebra • 2. Linear Equations And Inequalities • 15 questions available

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What is Solve inequalities using the Multiplication Property of Inequality?

Definition:
The Multiplication Property of Inequality states that multiplying both sides by a positive number keeps the inequality direction, but multiplying by a negative number reverses it. If $a > b$ and $c > 0$ , then $ac > bc$ ; if $c < 0$ , then $ac < bc$ .

Working:
For $x/3 \geq 2$ , multiply by positive 3: $x \geq 6$ . For $-x/2 < 4$ , multiply by -2 and reverse: $x > -8$ .

Example:
Solve $\frac{x}{4} < 3$ . Multiply by 4 (positive): $x < 12$ .

Reason:
This property handles fractions and is essential for solving inequalities where the variable is divided by a constant.

Easy

Medium

Hard

📝 All Solve inequalities using the Multiplication Property of Inequality MCQs

Q1. Which statement correctly describes what happens when both sides of an inequality are multiplied by a negative number?

A.The inequality symbol remains unchanged

B.The inequality symbol reverses direction ✅

C.Only the larger number changes

D.The inequality becomes an equation

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: When both sides of an inequality are multiplied or divided by a negative number, their order on the number line reverses. Therefore, the inequality symbol must also reverse to preserve the set of solutions.

Q2. Solve $-4x>20$ . Which value of $x$ satisfies the original inequality?

x=-6

✅

x=-5

x=4

x=6

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Dividing both sides by $-4$ requires reversing the inequality, giving $x<-5$ . Among the choices, only $x=-6$ is less than $-5$ , so it satisfies the original inequality.

Q3. A student solves $-3x\leq18$ and writes $x\leq-6$ . What is the most important correction?

A.Change

-6

to 6 only

B.Reverse the inequality to

x\geq-6

✅

C.Change

\leq

<

D.Multiply both sides by 3 without changing anything

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Dividing both sides by the negative coefficient $-3$ reverses the inequality. Thus $-3x\leq18$ becomes $x\geq-6$ . The student's numerical boundary is correct, but the direction is incorrect.

Q4. A temperature model gives $-2t<-14$ , where $t$ represents hours before a reference time. Which interpretation is mathematically consistent?

t<-7

t>-7

✅

t\leq-7

t\geq7

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Dividing by $-2$ reverses the inequality, producing $t>7$ , not $t>-7$ . Therefore, none of the listed choices is correct if the original inequality is exactly $-2t<-14$ . The correct solution is $t>7$ .

Q5. Which inequality has the same solution set as $x\geq-8$ ?

-2x\leq16

✅

2x\leq-16

-2x\geq16

4x\geq-32

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Multiplying $x\geq-8$ by the negative number $-2$ reverses the inequality, giving $-2x\leq16$ . This tests whether the learner understands both the numerical multiplication and the required reversal.

Q6. A delivery company models acceptable fuel efficiency with $0.75m>18$ , where $m$ is the number of miles traveled per unit of fuel. What conclusion follows?

m>13.5

m>24

✅

m<24

m<13.5

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: To isolate $m$ , divide both sides by the positive number 0.75. Because the multiplier is positive, the inequality direction stays unchanged, giving $m>18\div0.75=24$ .

Q7. A student claims that multiplying $x<-3$ by $-5$ gives $-5x<-15$ . Which response best evaluates the claim?

A.Correct, because multiplication never changes inequality direction

B.Incorrect, because multiplication by a negative reverses the inequality, giving

-5x>15

✅

C.Incorrect, because the result should be

-5x<-8

D.Correct, because both numbers become negative

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Multiplication by a negative number reverses the order of values. Starting with $x<-3$ and multiplying by $-5$ gives $-5x>15$ . The student's error is failing to reverse the inequality symbol.

Q8. A student solves $-\frac{1}{2}x>6$ by multiplying both sides by 2 and obtains $-x>12$ , then writes $x>12$ . What is the correct final result?

x>12

x<-12

✅

x>-6

x<6

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Multiplying by 2 first gives $-x>12$ . To isolate $x$ , multiply by $-1$ , which reverses the inequality and produces $x<-12$ . The student's final step incorrectly preserves the inequality direction.

Q9. On a number line, a graph has an open circle at -4 and shading extending to the right. Which inequality does the graph represent?

x>-4

✅

x\geq-4

x<-4

x\leq-4

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: An open circle means the endpoint is excluded, while shading to the right represents values greater than the endpoint. Therefore, the graph represents $x>-4$ , excluding $-4$ itself.

Q10. Two students solve $-6x\geq42$ . Student A obtains $x\leq-7$ , while Student B obtains $x\geq-7$ . Which evaluation is correct?

A.Both are correct because equality is included

B.Student A is correct because division by a negative reverses the inequality ✅

C.Student B is correct because division never changes direction

D.Neither is correct because

x=7

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Dividing $-6x\geq42$ by $-6$ reverses the inequality, producing $x\leq-7$ . Therefore Student A is correct. Student B demonstrates the common misconception that division leaves the direction unchanged.

Q11. A profit condition is represented by $-0.4p\leq-12$ , where $p$ is a production level. Which range of production levels satisfies the condition?

p\leq30

p\geq30

✅

p\leq-30

p\geq-30

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Dividing both sides by $-0.4$ reverses the inequality. Thus $p\geq(-12)/(-0.4)=30$ . The negative multiplier is the critical feature because ignoring the reversal would produce the wrong production range.

Q12. Which inequality produces a solution set extending leftward from -9 with a closed endpoint?

2x\geq-18

-2x\geq18

-2x\leq18

2x\leq-18

✅

💡 Difficulty: hard | ✅ Correct: D

📖 Explanation: A closed endpoint at -9 with shading left represents $x\leq-9$ . Dividing $2x\leq-18$ by the positive number 2 gives exactly $x\leq-9$ , without reversing the inequality.

Q13. A researcher requires a measurement $q$ to satisfy $-5q>-35$ . Another researcher argues that $q>7$ because both sides were divided by 5. Which conclusion is correct?

A.The argument is correct and

q>7

B.The correct solution is

q<-7

C.The correct solution is

q<7

✅

D.The correct solution is

q\geq7

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Dividing both sides by $-5$ , not 5, reverses the inequality. Therefore $-5q>-35$ becomes $q<7$ . The error comes from overlooking the negative coefficient attached to $q$ .

Q14. Find the value of $k$ for which the inequality $kx<-12$ has the solution $x>4$ .

k=3

k=-3

✅

k=-4

k=4

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: For $kx<-12$ to become $x>4$ , dividing by $k$ must reverse the inequality, so $k$ must be negative. Requiring the boundary to be 4 gives $-12/k=4$ , hence $k=-3$ .

Q15. A machine operates safely when $-\frac{3}{4}r\geq9$ , where $r$ is a control setting. Which statement correctly describes the safe settings?

r\geq-12

r\leq-12

✅

r\geq12

r\leq12

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Divide both sides by $-\frac{3}{4}$ . Because the divisor is negative, reverse the inequality: $r\leq9\div(-3/4)=-12$ . Thus settings at or below -12 satisfy the safety condition.

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