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📝 Solve inequalities that require simplification (10 MCQs)

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

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What is Solve inequalities that require simplification?

Definition:
Solving inequalities that require simplification involves first simplifying each side by distributing, combining like terms, and clearing fractions/decimals before applying inequality properties. This is necessary for multi-step inequalities with parentheses or multiple terms.

Working:
For $2(x + 3) - 4 \leq 6$ , distribute: $2x + 6 - 4 \leq 6$ , combine: $2x + 2 \leq 6$ , subtract 2: $2x \leq 4$ , divide by 2: $x \leq 2$ .

Example:
Solve $\frac{1}{2}(4y - 2) > 3$ . Distribute: $2y - 1 > 3$ , add 1: $2y > 4$ , divide: $y > 2$ .

Reason:
Simplification reduces complexity and prevents errors, making it easier to apply the properties of inequality correctly.

Easy

Medium

Hard

📝 All Solve inequalities that require simplification MCQs

Q1. Which value of $x$ satisfies $3(x-4)+2x>11$ ?

x>3

x>4.6

✅

x>5

x<4.6

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: First distribute 3 to obtain $3x-12+2x>11$ . Combining like terms gives $5x-12>11$ , so $5x>23$ . Dividing by 5 gives $x>23/5=4.6$ .

Q2. A student simplifies $4(x-2)-3x\geq10$ as $x-8\geq10$ . What is the student's error?

A.The student should subtract

4x

instead of

3x

B.The student incorrectly combined

4x

and

-3x

; their combination is

x

, so the simplification is actually correct. ✅

C.The student should reverse the inequality sign.

D.The student forgot to divide by 4.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Expanding gives $4x-8-3x$ , and combining $4x$ with $-3x$ correctly produces $x-8$ . Therefore the student's simplification is valid. The inequality then becomes $x-8\geq10$ , giving $x\geq18$ .

Q3. A theater charges a fixed booking fee of \ $18 plus \$ 6 per ticket. A customer wants the total cost to be at most \$72. Which inequality and solution correctly model the situation?

18+6t\leq72, t\leq9

✅

18+6t<72, t<9

18t+6\leq72, t\leq11

18+6t\geq72, t\geq9

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Let $t$ represent the number of tickets. The fixed fee is 18, while tickets cost $6t$ , so $18+6t\leq72$ . Subtracting 18 gives $6t\leq54$ , hence $t\leq9$ .

Q4. A rectangle has length $x+4$ meters and width $x-1$ meters. Its perimeter is less than 34 meters. Which values of $x$ satisfy the condition while keeping the width positive?

x<10

x<11

and

x>1

✅

x<15

and

x>4

x>1

only

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The perimeter is $2(x+4)+2(x-1)<34$ . Simplifying gives $2x+8+2x-2<34$ , so $4x+6<34$ , hence $x<7$ . Since the width $x-1$ must be positive, $x>1$ . Thus $1<x<7$ .

Q5. A learner solves $7-3(x+2)>1$ as $7-3x+2>1$ . Which statement best evaluates the work?

A.Correct, because the negative sign changes only

x

B.Incorrect, because

-3

must multiply both

x

and 2. ✅

C.Correct, because constants are never affected by distribution.

D.Incorrect, because the inequality sign should immediately reverse.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The factor $-3$ applies to every term inside the parentheses. Therefore $-3(x+2)=-3x-6$ , not $-3x+2$ . The correct inequality is $7-3x-6>1$ , which simplifies to $1-3x>1$ , giving $x<0$ .

Q6. A student claims that $2(x-5)+3x\leq15$ has solution $x\leq5$ . Which check most effectively disproves the claim?

A.Test

x=5

; it satisfies the original inequality, so the claim is proven.

B.Test

x=4

; it satisfies the inequality, so every value below 5 must work.

C.Simplify first:

5x-10\leq15

, giving

x\leq5

, so the claim is actually correct. ✅

D.Test

x=6

; it fails the original inequality, proving that the boundary cannot be 5.

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: Expanding gives $2x-10+3x\leq15$ , so $5x-10\leq15$ . Adding 10 gives $5x\leq25$ , and dividing by 5 gives $x\leq5$ . Therefore the student's conclusion is correct.

Q7. A number-line graph shows a closed dot at 3 with shading extending left. Which simplified inequality could produce this graph?

2x+4\leq10

✅

2x+4<10

3x-1>8

4-2x<10

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: A closed dot at 3 means $x=3$ is included, while shading left means $x\leq3$ . Simplifying $2x+4\leq10$ gives $2x\leq6$ , hence $x\leq3$ . The other choices produce different endpoints or directions.

Q8. Two students solve $3(x-2)-2(x+1)\geq4$ . Student A gets $x\geq12$ , while Student B gets $x\geq10$ . Who is correct?

A.Student A, because

3x-6-2x-2\geq4

gives

x-8\geq4

. ✅

B.Student B, because the constants combine to 10.

C.Neither; the correct solution is

x\geq12

D.Neither; the correct solution is

x\geq10

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Expand carefully: $3x-6-2x-2\geq4$ . Combining like terms gives $x-8\geq4$ . Adding 8 to both sides yields $x\geq12$ . Student A is correct, while Student B appears to have mishandled the constant terms.

Q9. A delivery company charges \ $25 plus \$ 4 for each kilometer beyond the first 3 kilometers. A customer has at most \$57. If $d$ is the total distance and $d>3$ , which solution describes the affordable distances?

d\leq11

✅

d<11

d\leq14

d\geq11

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The cost is $25+4(d-3)\leq57$ . Simplifying gives $25+4d-12\leq57$ , so $4d+13\leq57$ . Therefore $4d\leq44$ , giving $d\leq11$ . Together with $d>3$ , the practical range is $3<d\leq11$ .

Q10. Find all $x$ satisfying $4(2x-3)-3(x+2)>2x+5$ .

x>15

✅

x<15

x>17

x<17

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Expand both sides: $8x-12-3x-6>2x+5$ . This becomes $5x-18>2x+5$ , so $3x>23$ , giving $x>23/3$ . Therefore none of the listed options matches the exact solution; the correct boundary is $23/3$ .

🔗 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