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📝 Solve inequalities using the Division Property of Inequality (11 MCQs)

📖 From Digital SAT Algebra • 2. Linear Equations And Inequalities • 11 questions available

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What is Solve inequalities using the Division Property of Inequality?

Definition:
The Division Property of Inequality states that dividing both sides of an inequality by a positive number preserves the inequality direction. However, if you divide by a negative number, you must reverse the inequality sign. For $a < b$ and $c > 0$ , $a/c < b/c$ ; if $c < 0$ , $a/c > b/c$ .

Working:
For $6x \leq 24$ , divide by positive 6: $x \leq 4$ (no reversal). For $-4x > 20$ , divide by -4 and reverse: $x < -5$ .

Example:
Solve $8y \geq 40$ . Divide by 8 (positive): $y \geq 5$ .

Reason:
This property is crucial for isolating the variable when it has a coefficient, and the sign reversal rule is a key distinction from equations.

Easy

Medium

Hard

📝 All Solve inequalities using the Division Property of Inequality MCQs

Q1. Solve $-6x > 42$ . Which solution set correctly describes all possible values of $x$ ?

x > -7

x < -7

✅

x > 7

x < 7

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Dividing both sides by $-6$ requires reversing the inequality symbol because the divisor is negative. Thus $x > 42/(-6)$ would be incorrect; the correct result is $x < -7$ .

Q2. A student solves $15x \leq 90$ by dividing both sides by 15. Which statement best explains why the inequality symbol does not change?

A.Any division keeps the symbol unchanged

B.The divisor is positive, so the order of the numbers is preserved ✅

C.The variable is on the left side

D.The constant on the right side is positive

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Dividing an inequality by a positive number preserves the original order of the two sides. Therefore, dividing $15x \leq 90$ by 15 gives $x \leq 6$ without reversing the inequality symbol.

Q3. A delivery company allows a package to weigh at most 48 kg. If each identical container weighs 6 kg, which inequality represents the maximum number $n$ of containers that can be loaded?

n \geq 8

n < 8

n \leq 8

✅

n > 8

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The total weight is $6n$ , and the requirement is $6n \leq 48$ . Dividing both sides by the positive number 6 preserves the inequality, giving $n \leq 8$ . Thus at most eight containers can be loaded.

Q4. A student claims that solving $-9x \leq 63$ gives $x \leq -7$ . What is the best evaluation of the student's reasoning?

A.Correct, because division never changes inequality symbols

B.Correct, because both numbers are divisible by 9

C.Incorrect, because dividing by a negative number reverses the symbol, giving

x \geq -7

✅

D.Incorrect, because the answer should be

x \geq 7

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The student correctly divides the magnitudes but fails to reverse the inequality symbol. Since $-9$ is negative, dividing both sides by $-9$ changes $\leq$ to $\geq$ , producing $x \geq -7$ .

Q5. A number-line graph has a closed point at -4 and is shaded to the left. Which inequality could have produced this graph after dividing by a negative coefficient?

x \geq -4

x \leq -4

✅

x > -4

x < 4

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: A closed point means $-4$ is included, while shading left represents values less than $-4$ . Therefore, the graph represents $x \leq -4$ , which could result when a negative coefficient is divided out and the inequality reverses.

Q6. A savings plan requires $4m-12 \geq 20$ , where $m$ represents monthly deposits. Which value of $m$ is the smallest integer that satisfies the requirement?

A.6

B.7

C.8 ✅

D.9

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: First add 12 to both sides to obtain $4m \geq 32$ . Dividing by the positive number 4 gives $m \geq 8$ . Therefore, the smallest integer deposit count represented by $m$ that satisfies the inequality is 8.

Q7. Which inequality has the same solution set as $-12y > 36$ ?

y > -3

y < -3

✅

y > 3

y < 3

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Because $-12$ is negative, dividing both sides by it reverses the inequality. Thus $-12y > 36$ becomes $y < 36/(-12)$ , so the solution is $y < -3$ .

Q8. A teacher asks two students to solve $-5x < 25$ . Student A writes $x < -5$ , while Student B writes $x > -5$ . Which student is correct, and why?

A.Student A, because dividing by 5 keeps the sign

B.Student B, because dividing by a negative reverses the inequality ✅

C.Student A, because the coefficient is negative

D.Both students, because both sets contain -5

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Student B is correct. Dividing by $-5$ reverses the inequality from $<$ to $>$ . Therefore, $-5x < 25$ becomes $x > -5$ , not $x < -5$ .

Q9. A number-line graph represents $x > 3$ . Which original inequality could produce this graph when the variable term is divided by $-2$ ?

-2x > -6

-2x < -6

✅

2x > 6

2x < -6

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Dividing $-2x < -6$ by the negative number $-2$ reverses the inequality, producing $x > 3$ . The open endpoint at 3 is consistent because the original inequality is strict.

Q10. A machine operates safely when $-8t \geq -56$ , where $t$ is a temperature-related parameter. A technician concludes $t \geq 7$ . Is the conclusion valid?

A.Yes, because dividing by 8 keeps the inequality

B.No, the correct solution is

t \leq 7

✅

C.No, the correct solution is

t < -7

D.Yes, because both sides are negative

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Dividing $-8t \geq -56$ by the negative number $-8$ reverses the inequality. This gives $t \leq 7$ . The technician incorrectly retained the original direction, so the stated conclusion is invalid.

Q11. Suppose $a$ is a negative number and $ax \geq b$ . Without knowing the exact values of $a$ and $b$ , which transformation correctly isolates $x$ ?

x \geq b/a

x \leq b/a

✅

x > b/a

x < b/a

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Because $a$ is explicitly negative, dividing both sides of $ax \geq b$ by $a$ reverses the inequality. Therefore, the equivalent inequality is $x \leq b/a$ , regardless of the particular negative value of $a$ .

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