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📝 How to Solve Linear Inequalities (10 MCQs)

📖 From Digital SAT Algebra • 2. Linear Equations And Inequalities • 10 questions available

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What is How to Solve Linear Inequalities?

Definition:
Solving linear inequalities is similar to solving linear equations, but with inequality symbols ( $<, >, \leq, \geq$ ). The goal is to isolate the variable on one side. However, a key difference is that multiplying or dividing by a negative number reverses the inequality sign.

Working:
For $3x - 5 > 10$ , add 5: $3x > 15$ , divide by 3: $x > 5$ . For $-2x \leq 6$ , divide by -2 and reverse the sign: $x \geq -3$ . The solution set is often represented on a number line or in interval notation.

Example:
Solve $4y + 2 < 18$ . Subtract 2: $4y < 16$ , divide by 4: $y < 4$ . This means all numbers less than 4 are solutions.

Reason:
Inequalities are used to describe ranges, limits, and constraints in real-life scenarios like budgets, speed limits, and minimum requirements.

Easy

Medium

Hard

📝 All How to Solve Linear Inequalities MCQs

Q1. Which value of $x$ satisfies the inequality $4x-7>13$ ?

x=4

x=5

x=6

✅

x=7

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: Add 7 to both sides to obtain $4x>20$ , then divide by the positive number 4 without changing the inequality direction. This gives $x>5$ , so among the choices only $x=6$ and $x=7$ satisfy it; therefore option C is correct.

Q2. A student solves $-3x+8\leq20$ and writes $x\leq-4$ . What is the most important error in the student's reasoning?

A.The student should add 8 instead of subtracting 8.

B.The student forgot to divide by 3.

C.The inequality direction should reverse after division by

-3

. ✅

D.The constant 20 should become negative.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Subtracting 8 gives $-3x\leq12$ . Dividing by $-3$ requires reversing the inequality symbol, producing $x\geq-4$ . The student's numerical boundary is correct, but the direction is reversed incorrectly, which changes the entire solution set.

Q3. A delivery service charges a fixed fee of 6 dollars plus 2.50 dollars per mile. A customer has at most 31 dollars. Which inequality and solution correctly represent the greatest possible distance $m$ ?

6+2.50m\geq31, m\geq10

6+2.50m\leq31, m\leq10

✅

2.50m-6\leq31, m\leq14.8

6+31m\leq2.50, m\leq0

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The phrase 'at most' means the total cost cannot exceed 31, so $6+2.50m\leq31$ . Subtracting 6 gives $2.50m\leq25$ , and dividing by 2.50 gives $m\leq10$ . Thus the customer can travel no more than 10 miles.

Q4. A teacher claims that $5-2x>17$ gives $x>-6$ . Which explanation best evaluates the claim?

A.Correct, because subtracting 5 gives

-2x>12

B.Incorrect, because dividing by

-2

reverses the inequality, giving

x<-6

. ✅

C.Correct, because negative coefficients never affect inequality signs.

D.Incorrect, because the answer must be

x>-12

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Subtracting 5 from both sides produces $-2x>12$ . Dividing by $-2$ changes the direction from greater than to less than, giving $x<-6$ . The student's proposed $x>-6$ includes values that do not satisfy the original inequality.

Q5. A graph on a number line shows an open circle at 3 with shading extending to the left. Which inequality does the graph represent?

x>3

x\geq3

x<3

✅

x\leq3

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: An open circle means the boundary value 3 is excluded from the solution. Shading to the left represents numbers smaller than 3. Therefore the graph corresponds to $x<3$ , rather than either inequality that includes 3 or shades toward larger values.

Q6. A school club has 120 dollars and must reserve 30 dollars for supplies. Each membership packet costs 15 dollars. If $p$ packets are purchased, which solution represents the maximum number of packets?

p\leq6

✅

p\geq6

p\leq8

p\geq8

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The available amount for packets is $120-30=90$ dollars. Therefore $15p\leq90$ . Dividing by 15 gives $p\leq6$ . Since packets are counted in whole numbers, six is the maximum possible purchase while still preserving the required 30 dollars.

Q7. A student solves $2(x-4)<3x+5$ as follows: $2x-8<3x+5$ , then $-13<x$ , so $x>-13$ . What went wrong?

A.The distributive property was applied incorrectly.

B.The student should divide by 2 before expanding.

C.The inequality should reverse when subtracting

3x

, giving

x>-13

, so nothing is wrong.

D.The student made an arithmetic error when combining constants. ✅

💡 Difficulty: hard | ✅ Correct: D

📖 Explanation: The expansion $2(x-4)=2x-8$ is correct. Subtracting $3x$ gives $-x-8<5$ , then adding 8 gives $-x<13$ . Dividing by $-1$ reverses the inequality, producing $x>-13$ . Therefore the student's final answer is actually correct, so the claim that an error occurred is false.

Q8. Which inequality has the same solution set as $7-3x\geq16$ ?

x\geq-3

x\leq-3

✅

x\geq3

x\leq3

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Subtracting 7 gives $-3x\geq9$ . Dividing by $-3$ reverses the inequality, resulting in $x\leq-3$ . The negative coefficient is the key feature because failing to reverse the inequality would produce the opposite and incorrect solution set.

Q9. A number $x$ must satisfy both $x>2$ and $3x-4\leq11$ . Which description gives all possible values of $x$ ?

x\leq2

x>5

2<x\leq5

✅

2\leq x<5

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Solving $3x-4\leq11$ gives $3x\leq15$ , so $x\leq5$ . Combining this with $x>2$ requires values greater than 2 but no greater than 5. Thus the intersection of the two solution sets is $2<x\leq5$ .

Q10. A student says the inequality $4-5x<19$ has solution $x>-3$ . Another student says it has solution $x<-3$ . Which conclusion is correct, and why?

A.The first student is correct because the coefficient of

x

is positive after simplification. ✅

B.The second student is correct because dividing by a negative coefficient reverses the inequality.

C.Both are correct because inequalities can have two solution intervals.

D.Neither is correct; the solution is

x<3

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Subtracting 4 gives $-5x<15$ . Dividing by $-5$ reverses the inequality, producing $x>-3$ . Therefore the first student's answer is actually correct, while the second student's answer incorrectly ignores the reversal caused by dividing by a negative number.

🔗 Related Topics (MCQs)

📝 Subtraction and Addition Properties of Equality MCQs 📝 How to verify a solution to an equation MCQs 📝 Subtraction Property of Equality MCQs 📝 Addition Property of Equality MCQs 📝 Solving equations with simplification MCQs

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