← Back to 2. Linear Equations And Inequalities

📝 Solve inequalities using the Addition Property of Inequality (12 MCQs)

📖 From Digital SAT Algebra • 2. Linear Equations And Inequalities • 12 questions available

←Previous Topic:Solve inequalities using the Subtraction Property of Inequality📝 Solve inequalities using the Addition Property of InequalitySolve inequalities using the Division Property of Inequality:Next Topic→

What is Solve inequalities using the Addition Property of Inequality?

Definition:
The Addition Property of Inequality states that adding the same number to both sides of an inequality maintains the inequality direction. If $a > b$ , then $a + c > b + c$ . This is used to eliminate subtracted constants.

Working:
For $y - 3 < 5$ , add 3 to both sides: $y - 3 + 3 < 5 + 3$ , giving $y < 8$ . The direction remains the same.

Example:
Solve $p - 2 \geq -6$ . Add 2: $p \geq -4$ . The solution includes all numbers greater than or equal to -4.

Reason:
This property works identically to its equality counterpart, making it intuitive for students who have learned solving equations.

Easy

Medium

Hard

📝 All Solve inequalities using the Addition Property of Inequality MCQs

Q1. Which inequality is equivalent to $x-7>12$ after using the Addition Property of Inequality correctly?

x>5

x>19

✅

x<19

x<5

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Adding 7 to both sides preserves the direction of the inequality because the same quantity is added to each side. Thus $x-7+7>12+7$ , giving $x>19$ .

Q2. Which value of $k$ makes the inequality $k-15\leq-4$ equivalent to $k\leq11$ ?

A.4

B.11

C.15 ✅

D.19

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: To remove the $-15$ , add 15 to both sides. This produces $k\leq-4+15$ , so $k\leq11$ . Therefore, the value being added is 15.

Q3. Consider $y+6\geq14$ . A student subtracts 6 from both sides and obtains $y\geq20$ . What is the best diagnosis of the error?

A.The inequality should reverse when subtracting 6.

B.The student should add 6 instead, obtaining

y\geq20

C.The student used the wrong operation; subtracting 6 gives

y\geq8

. ✅

D.The variable must be divided by 6 before solving.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Subtracting 6 from both sides is appropriate because it cancels the positive 6 attached to $y$ . The calculation gives $y+6-6\geq14-6$ , so $y\geq8$ , not 20.

Q4. Which inequality has the same solution set as $m+18<7$ ?

m<-11

✅

m<25

m>11

m>-11

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Subtracting 18 from both sides isolates $m$ : $m+18-18<7-18$ . Therefore $m<-11$ . The inequality direction stays the same because subtraction is performed equally on both sides.

Q5. A temperature model gives $T-12\geq5$ , where $T$ represents the temperature in degrees. Which interpretation correctly describes the solution?

A.The temperature must be at least 7°

B.The temperature must be at least 17° ✅

C.The temperature must be less than 17°

D.The temperature must be greater than 5°

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Adding 12 to both sides gives $T\geq17$ . Because the inequality includes equality, a temperature of exactly 17° is allowed, as well as every temperature above 17°.

Q6. A savings account is modeled by $S-250>600$ , where $S$ is the amount currently in the account. What must be true about $S$ ?

S>350

S>850

✅

S<850

S<350

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The model subtracts 250 from the account amount. Adding 250 to both sides gives $S>600+250$ , or $S>850$ . Thus the account must contain more than 850 units of currency.

Q7. Two students solve $p-24\leq10$ . Student A obtains $p\leq34$ , while Student B obtains $p\leq-14$ . Which comparison is correct?

A.Student A is correct because 24 should be added to both sides. ✅

B.Student B is correct because subtracting 24 reverses the inequality.

C.Both are correct because either direction can represent the same solution.

D.Neither is correct because inequalities cannot contain negative values.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Adding 24 to both sides cancels the $-24$ , giving $p\leq10+24=34$ . Student B incorrectly subtracts 24 again, moving farther from isolating the variable rather than canceling its existing term.

Q8. A number-line graph shows an open circle at -3 with shading extending to the right. Which inequality matches the graph?

x\geq-3

x>-3

✅

x\leq-3

x<-3

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: An open circle means the boundary value is excluded, while shading to the right represents values greater than the boundary. Therefore, the graph represents $x>-3$ , not $x\geq-3$ .

Q9. A delivery company requires a driver to complete more than 40 deliveries. If the model is $d-12>28$ , where $d$ is the total number of deliveries, which conclusion follows?

d>16

d>28

d>40

✅

d>52

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Adding 12 to both sides gives $d>28+12$ , so $d>40$ . This agrees with the stated requirement that the driver must complete more than 40 deliveries, showing the algebraic solution has a meaningful interpretation.

Q10. A student transforms $r+5<2$ into $r<-3$ , then says the inequality should become $r>-3$ because subtraction was used. What is wrong with the student's final statement?

A.Subtraction always reverses an inequality.

B.Only multiplication reverses an inequality.

C.Subtracting the same number from both sides does not reverse the inequality. ✅

D.The number 5 should be multiplied instead of subtracted.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Subtracting 5 from both sides gives $r+5-5<2-5$ , so $r<-3$ . The inequality direction remains unchanged. Reversal occurs when multiplying or dividing by a negative number, not when adding or subtracting.

Q11. A project has a maximum time limit represented by $t+18\leq50$ . Another project has a requirement $q-10>25$ . Which pair correctly describes the allowable values?

t\leq68, q>15

t\leq32, q>35

✅

t\geq32, q<35

t\leq32, q>15

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: For the first inequality, subtract 18 to obtain $t\leq32$ . For the second, add 10 to obtain $q>35$ . Both operations preserve the inequality directions, so option B correctly combines the two solution sets.

Q12. An inequality has the form $x+a>b$ . A student says that the solution is always $x>b-a$ , regardless of whether $a$ is positive or negative. Is the statement valid?

A.Yes, because addition always produces the same result.

B.No, because the solution should be

x>b+a

C.Yes, because subtracting

a

from both sides gives

x>b-a

, even when

a

is negative. ✅

D.No, because the inequality direction must always reverse when

a

is negative.

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The statement is valid because subtracting $a$ from both sides isolates $x$ , giving $x>b-a$ . If $a$ itself is negative, subtracting a negative number is equivalent to adding its positive opposite, but the inequality direction still does not reverse.

🔗 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