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📝 Solve inequalities using the Subtraction Property of Inequality (13 MCQs)

📖 From Digital SAT Algebra • 2. Linear Equations And Inequalities • 13 questions available

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What is Solve inequalities using the Subtraction Property of Inequality?

Definition:
The Subtraction Property of Inequality states that subtracting the same number from both sides of an inequality preserves the inequality direction. For $a < b$ , $a - c < b - c$ . This is used to eliminate added constants on the variable side.

Working:
For $x + 7 > 12$ , subtract 7 from both sides: $x + 7 - 7 > 12 - 7$ , giving $x > 5$ . No sign reversal occurs.

Example:
Solve $m + 4 \leq 10$ . Subtract 4: $m \leq 6$ . The solution is all numbers less than or equal to 6.

Reason:
This property allows us to isolate the variable by removing constants, just like in equations, but without changing the inequality.

Easy

Medium

Hard

📝 All Solve inequalities using the Subtraction Property of Inequality MCQs

Q1. Solve $x+7>15$ by subtracting the same number from both sides. Which solution set is correct?

x>8

✅

x>22

x<8

x\geq8

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Subtracting 7 from both sides preserves the direction of the inequality because the same quantity is removed from each side. Thus $x+7>15$ becomes $x>8$ , so all values greater than 8 satisfy the inequality.

Q2. A student solves $y+12\leq20$ and writes $y\leq32$ . Which correction best explains the error?

A.Add 12 to both sides instead of subtracting it

B.Subtract 12 from both sides, giving

y\leq8

✅

C.Reverse the inequality sign, giving

y\geq8

D.Divide both sides by 12, giving

y\leq\frac{20}{12}

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The student incorrectly added 12 instead of removing it from the left side. Subtracting 12 from both sides gives $y\leq8$ . Since subtraction does not reverse an inequality, the sign remains $\leq$ .

Q3. A temperature must remain below 18°C. If the temperature is modeled by $T+5<18$ , which condition describes the allowable values of $T$ ?

T<13

✅

T>13

T<23

T>23

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: To isolate $T$ , subtract 5 from both sides: $T+5<18$ becomes $T<13$ . Therefore, any temperature strictly below 13°C meets the stated requirement.

Q4. Which inequality is equivalent to $3+x\geq11$ , and why?

x\geq8

, because 3 is subtracted from both sides ✅

x\geq14

, because 3 is added to both sides

x\leq8

, because the inequality reverses

x\leq14

, because subtraction reverses the inequality

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Subtracting 3 from both sides isolates $x$ : $3+x\geq11$ becomes $x\geq8$ . The inequality direction does not change because subtracting the same quantity from both sides preserves the original ordering.

Q5. A club has already collected 15 points and needs more than 40 points in total. If $p+15>40$ , what is the minimum whole-number value of $p$ ?

A.24

B.25

C.26 ✅

D.55

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Subtracting 15 gives $p>25$ . Because $p$ represents a whole number of additional points, the smallest possible value strictly greater than 25 is 26. This distinguishes an inequality boundary from its first integer solution.

Q6. A student claims that $a-9<4$ becomes $a<-5$ after subtracting 9 from both sides. What is the correct reasoning?

A.Add 9 to both sides, obtaining

a<13

✅

B.Subtract 9 again, obtaining

a<-5

C.Reverse the sign, obtaining

a>13

D.Multiply by

-1

, obtaining

a>5

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The expression $a-9$ already has 9 subtracted from $a$ . To isolate $a$ , add 9 to both sides: $a-9<4$ becomes $a<13$ . The student's second subtraction moves the variable farther from isolation.

Q7. A number line represents all values to the left of 10, with 10 shown by an open circle. Which inequality matches the graph?

x\leq10

x<10

✅

x>10

x\geq10

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: An open circle means the boundary value 10 is excluded, while shading to the left represents values smaller than 10. Therefore, the graph corresponds precisely to $x<10$ , not $x\leq10$ .

Q8. A number-line graph shows a closed circle at 6 and shading to the left. Which original inequality could produce this solution after subtracting 4 from both sides?

x+4\leq10

✅

x+4<10

x-4\leq10

x+4\geq10

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The graph represents $x\leq6$ . Adding 4 to both sides reverses the subtraction step, giving $x+4\leq10$ . The closed circle confirms that 6 is included, so a strict inequality would be incorrect.

Q9. Two students solve $m+18>30$ . Student A subtracts 18 from both sides. Student B subtracts 30 from both sides. Who obtains an equivalent inequality?

A.Only Student A, because

m>12

isolates the variable

B.Only Student B, because

m-12>0

is simpler

C.Both, because both operations preserve equivalence ✅

D.Neither, because subtraction always reverses

>

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Student A obtains $m>12$ . Student B obtains $m-12>0$ , which is also equivalent to $m>12$ after rearranging. Subtracting the same quantity from both sides preserves the inequality, so both methods are valid.

Q10. A delivery service charges a fixed fee of 8 dollars plus an amount $c$ for the package. The total must be at most 35 dollars. Which inequality describes the allowable package charge?

c\leq27

✅

c<27

c\geq27

c\leq43

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The cost condition is $c+8\leq35$ . Subtracting 8 from both sides gives $c\leq27$ . The phrase 'at most' includes the boundary value, so 27 is allowed and the correct symbol is $\leq$ .

Q11. Suppose $q+14\geq22$ . A student says $q\geq36$ because the numbers 14 and 22 should be combined. Which statement best evaluates the student's approach?

A.Correct, because numbers on opposite sides should be added

B.Incorrect; subtract 14 from both sides to get

q\geq8

✅

C.Incorrect; subtract 22 from both sides to get

q\geq-8

D.Correct, because

14+22=36

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The goal is to isolate $q$ , not combine unrelated constants. Subtracting 14 from both sides gives $q\geq8$ . The student's 36 comes from an operation that changes the equation's meaning rather than preserving equivalence.

Q12. A solution must satisfy $r+6>2$ and $r+10\leq15$ . Which interval contains exactly the values satisfying both conditions?

r>4

r\leq5

4<r\leq5

✅

r<4

r>5

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Subtracting 6 from the first inequality gives $r>-4$ , while subtracting 10 from the second gives $r\leq5$ . Combining both conditions produces $-4<r\leq5$ . This requires solving and intersecting two inequalities rather than treating them separately.

Q13. For the inequality $x+9<16$ , a student obtains $x<7$ . Another student obtains $x<25$ . Without testing many values, which result must be correct and why?

x<25

, because addition requires increasing 16 by 9

x<7

, because subtracting 9 reverses the inequality

C.Both are correct because

7<25

x<7

, because subtracting 9 isolates

x

without changing the sign ✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: Subtracting 9 from both sides gives $x<7$ . The inequality direction stays unchanged because the same quantity is subtracted from both sides. The result $x<25$ comes from incorrectly adding 9 instead of removing it.

🔗 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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