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📝 Reverse the inequality sign when multiplying or dividing by a negative number (15 MCQs)

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

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What is Reverse the inequality sign when multiplying or dividing by a negative number?

Definition:
Reversing the inequality sign when multiplying or dividing by a negative number is a critical rule: whenever you multiply or divide both sides of an inequality by a negative value, you must flip the direction of the inequality symbol (e.g., $<$ becomes $>$ , $\leq$ becomes $\geq$ ).

Working:
For $-3x \leq 9$ , divide by -3: $x \geq -3$ (sign reversed). For $-2y > 10$ , divide by -2: $y < -5$ (sign reversed).

Example:
Solve $-5m \geq 20$ . Divide by -5 and reverse: $m \leq -4$ . Check: if $m = -5$ , $-5(-5) = 25 \geq 20$ , true.

Reason:
This rule maintains the logical truth of the inequality because multiplying by a negative flips the order on the number line.

Easy

Medium

Hard

📝 All Reverse the inequality sign when multiplying or dividing by a negative number MCQs

Q1. Which value of $x$ satisfies $-4x > 20$ ?

x > -5

x < -5

✅

x > 5

x < 5

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Dividing both sides of $-4x > 20$ by $-4$ requires reversing the inequality because the divisor is negative. Therefore, $x < -5$ , making option B the correct solution.

Q2. A student solves $-3x \leq 18$ and writes $x \leq -6$ . Which statement best evaluates the student's work?

A.The solution is correct because only the number changes.

B.The inequality should remain unchanged because division is always reversible.

C.The inequality should reverse, giving

x \geq -6

. ✅

D.The answer should be

x \geq 6

because both signs are negative.

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: The student correctly divides 18 by $-3$ to obtain $-6$ , but dividing by a negative number reverses the inequality. Thus the correct result is $x \geq -6$ , not $x \leq -6$ .

Q3. Which inequality is equivalent to $x > -7$ after multiplying both sides by $-2$ ?

-2x > 14

-2x < 14

✅

-2x > -14

-2x < -14

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Multiplying an inequality by a negative number reverses its direction. Starting with $x > -7$ , multiplication by $-2$ gives $-2x < 14$ . The reversal is essential for preserving the same solution set.

Q4. A temperature model gives $T = -5x + 30$ , where $x$ represents hours after midnight. For which values of $x$ is the temperature below 10 degrees?

x < 4

x > 4

✅

x < -4

x > -8

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: To find when $T<10$ , solve $-5x+30<10$ . Subtracting 30 gives $-5x<-20$ . Dividing by $-5$ reverses the sign, producing $x>4$ .

Q5. A shipping company uses the model $C=50-4d$ , where $d$ is a discount percentage expressed as a number. A customer wants $C\leq22$ . What condition must $d$ satisfy?

d\leq7

d\geq7

✅

d\leq28

d\geq28

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Set $50-4d\leq22$ . Subtracting 50 gives $-4d\leq-28$ . Dividing by the negative coefficient reverses the inequality, so $d\geq7$ . This correctly models the required discount.

Q6. A student claims that multiplying $-2<5$ by $-3$ gives $-6<-15$ . What is the strongest reason the result is incorrect?

A.The multiplication should use 3, not

-3

B.The inequality reverses when both sides are multiplied by a negative number. ✅

C.Only the larger number should be multiplied.

D.Multiplication never changes an inequality.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Multiplying both sides of an inequality by a negative number reverses their order. Therefore, multiplying $-2<5$ by $-3$ produces $6>-15$ , not $-6<-15$ .

Q7. A number-line graph shows an open circle at $-3$ with shading extending to the right. Which inequality could represent the graph?

x<-3

x\leq-3

x>-3

✅

x\geq-3

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: An open circle means $-3$ is excluded, while shading to the right represents numbers greater than $-3$ . Therefore, the graph represents $x>-3$ , making option C correct.

Q8. A number-line graph represents $x\leq4$ . Which inequality results when both sides are multiplied by $-3$ ?

-3x\leq-12

-3x\geq-12

✅

-3x\geq12

-3x\leq12

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Because $x\leq4$ , multiplying both sides by $-3$ reverses the inequality. The right side becomes $-12$ , so the equivalent inequality is $-3x\geq-12$ .

Q9. Which inequality has the same solution set as $-6x+9\geq45$ ?

x\geq-6

x\leq-6

✅

x\geq9

x\leq9

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Subtract 9 from both sides to obtain $-6x\geq36$ . Dividing by $-6$ reverses the inequality, giving $x\leq-6$ . The negative coefficient is the key detail determining the direction.

Q10. Two students solve $-8x<24$ . Student A obtains $x<-3$ , while Student B obtains $x>-3$ . Who is correct and why?

A.Student A, because division preserves the sign.

B.Student B, because division by

-8

reverses the inequality. ✅

C.Both are correct because the inequalities are equivalent.

D.Neither is correct because

x=3

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Dividing $-8x<24$ by $-8$ changes the direction of the inequality. Thus $x>-3$ . Student A made the common error of keeping the original inequality direction.

Q11. A learner starts with $-2x+7>15$ , subtracts 7, and gets $-2x>8$ . They then write $x>-4$ . What correction is needed?

A.No correction; the answer is correct.

B.The learner should divide by 2, giving

x>4

C.The learner should divide by

-2

and reverse the sign, giving

x<-4

. ✅

D.The learner should add 2, giving

x<-6

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: After subtracting 7, the inequality is $-2x>8$ . Dividing by $-2$ changes the inequality direction, so $x<-4$ . The numerical division is correct, but the sign reversal was omitted.

Q12. A graph shades all numbers to the left of $-2$ , including $-2$ . Which transformed inequality could produce this graph after dividing by a negative number?

-4x\leq8

-4x\geq8

✅

4x\leq-8

4x\geq-8

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Solving $-4x\geq8$ requires division by $-4$ , which reverses the inequality and gives $x\leq-2$ . This matches a graph shaded left with a closed endpoint at $-2$ .

Q13. A company models its daily profit as $P=-12x+240$ . The company needs profit to be at least 96. What is the greatest value of $x$ that satisfies this requirement?

x\leq8

x\geq8

x\leq12

✅

x\geq12

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The requirement is $-12x+240\geq96$ . Subtracting 240 gives $-12x\geq-144$ . Dividing by $-12$ reverses the inequality, giving $x\leq12$ . Therefore, the greatest possible value is 12.

Q14. Which reasoning correctly explains why the inequality direction changes when both sides are multiplied by a negative number?

A.Negative multiplication changes every number into zero.

B.Negative multiplication reverses the order of numbers on the number line. ✅

C.Negative multiplication always makes the inequality false.

D.Negative multiplication changes an inequality into an equation.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Multiplication by a negative number reflects values across zero on the number line, reversing their order. Because the relative order changes, the inequality symbol must also reverse to maintain an equivalent statement.

Q15. Find all real numbers $x$ satisfying $-\frac{x}{3}>5$ , then identify the correct description of the solution set.

x>15

x<-15

✅

x>-15

x<15

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Multiplying both sides by 3 first gives $-x>15$ . Multiplying by $-1$ then reverses the inequality, producing $x<-15$ . The two-step reasoning requires recognizing a second sign reversal.

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