📝 Subtract Integers in algebraic expressions (28 MCQs)

📖 From Digital SAT Algebra • 1. Basics of Algebra • 28 questions available

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What is Subtract Integers in algebraic expressions?

Definition:
Subtracting integers in algebraic expressions involves finding the difference between signed numbers, which is equivalent to adding the opposite (or additive inverse) of the integer being subtracted, and this rule applies to both constants and coefficients in variable expressions.

Working:
To subtract, change the subtraction sign to addition and change the sign of the number after the subtraction sign (e.g., $a - b = a + (-b)$ ), then apply the addition rules for integers; in expressions, simplify variable and constant terms separately.

Example:
Simplify $3a - 7 - (-2a) - 4$ .
Solution: Rewrite as $3a + (-7) + 2a + (-4)$ ; combine $3a+2a=5a$ , constants $-7-4=-11$ ; result $5a - 11$ .

Reason:
This concept is fundamental for solving equations where variables are on both sides, and it helps in understanding inverse operations, which are key to isolating variables and finding solutions in algebra.

Easy

Medium

Hard

📝 All Subtract Integers in algebraic expressions MCQs

Q1. A temperature is $4^\circ\text{C}$ at noon and $9^\circ\text{C}$ below zero at midnight. Which expression and result correctly represent the temperature change from noon to midnight?

4-(-9)=13

4-9=-5

-9-4=-13

✅

-9-(-4)=-5

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The final temperature is $-9^\circ\text{C}$ and the initial temperature is $4^\circ\text{C}$ , so the change is final minus initial: $-9-4=-13$ . The negative result correctly indicates a decrease of $13^\circ\text{C}$ .

Q2. A student claims that subtracting an integer always makes the answer smaller because subtraction means taking something away. Which example most clearly disproves the claim?

7-3=4

-5-2=-7

6-(-4)=10

✅

-8-1=-9

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Subtracting a negative integer is equivalent to adding its opposite. Therefore, $6-(-4)=6+4=10$ , which is greater than the starting value. This demonstrates why subtraction does not always produce a smaller number.

Q3. Two players track their scores relative to zero. Player A has $-12$ points and gains $7$ points, while Player B starts at $5$ points and loses $14$ points. What is the difference between their final scores?

A.-2

B.-4 ✅

C.-6

D.-10

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Player A finishes at $-12+7=-5$ , while Player B finishes at $5-14=-9$ . The difference is $-5-(-9)=4$ . The key step is comparing the final values rather than the amounts gained or lost.

Q4. A student simplifies $-6-(-11)$ as $-6-11=-17$ . What mistake did the student make, and what is the correct result?

A.They ignored the second negative sign; the result is

5

✅

B.They changed both signs; the result is

-5

C.They subtracted absolute values; the result is

17

D.They treated subtraction as addition; the result is

-17

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The expression contains subtraction of a negative integer. Rewriting it as $-6+11$ gives $5$ . The student's error was treating the negative sign inside the parentheses as though it were irrelevant.

Q5. A number-line graph shows a point moving from $3$ to $-5$ . The movement is represented by the expression $-5-3$ . Which interpretation is correct?

A.The point moved

8

units to the right

B.The point moved

8

units to the left ✅

C.The point moved

2

units to the left

D.The point moved

2

units to the right

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The displacement is final position minus initial position, so $-5-3=-8$ . A negative displacement means movement to the left on a number line, and the magnitude $8$ gives the distance traveled.

Q6. A bank account has a balance of $-\18\). After a transaction, the balance becomes $-\7$ . Which subtraction expression represents the change in balance, and why?

-18-(-7)=-11

-7-(-18)=11

✅

18-7=11

-18-7=-25

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The change is calculated as final balance minus initial balance: $-7-(-18)=11$ . The account increased by $11$ dollars even though both balances were negative, illustrating that subtracting a negative can produce a positive change.

Q7. Suppose $x$ is an integer and $x-(-6)=2$ . A student argues that $x=-4$ because the signs should simply be combined. Which reasoning correctly determines $x$ ?

A.Rewrite as

x+6=2

, so

x=-4

✅

B.Rewrite as

x-6=2

, so

x=8

C.Rewrite as

x+6=2

, so

x=8

D.Rewrite as

x-6=2

, so

x=-4

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Subtracting $(-6)$ is equivalent to adding $6$ , giving $x+6=2$ . Subtracting $6$ from both sides yields $x=-4$ . The solution requires interpreting the two signs rather than combining them mechanically.

Q8. Using $+$ counters for positive integers and $-$ counters for negative integers, a model contains $6$ positive counters and $4$ negative counters. If $3$ negative counters are removed, what integer does the remaining model represent?

A.-1

B.-3

C.-5 ✅

D.-7

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The original model represents $6+(-4)=2$ . Removing $3$ negative counters changes the model to $6+(-1)=5$ . Thus the subtraction $2-(-3)$ is modeled by removing three negative counters, giving $5$ .

Q9. A counter model represents $4$ as four positive counters. A student wants to model $4-6$ . What must be added before six positive counters can be removed, and what result should the completed model show?

A.Add

2

negative counters and remove six positive counters; result

=-2

✅

B.Add

6

negative counters and remove six positive counters; result

=6

C.Add

2

positive counters and remove six negative counters; result

=2

D.Add

6

zero pairs and remove six positive counters; result

=2

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: There are only four positive counters, so six cannot be removed directly. Adding two zero pairs creates six positive and two negative counters without changing the value. Removing six positive counters leaves two negative counters, representing $-2$ .

Q10. A model contains $3$ positive counters and $7$ negative counters. A student says that subtracting $4$ negative counters means removing four positive counters instead. Which statement best evaluates the student's reasoning?

A.Correct, because subtraction always removes positive counters

B.Correct, because negative counters have no effect on the value

C.Incorrect, because subtracting

4

negative counters requires removing four negative counters ✅

D.Incorrect, because negative counters can never be removed

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: When modeling subtraction, the counters being removed must match the sign of the integer being subtracted. Therefore, subtracting $-4$ means removing four negative counters. If they are unavailable, zero pairs can first be added.

Q11. A counter model represents $-3$ using three negative counters. To model $-3-(-5)$ , a student adds five zero pairs and then removes five negative counters. What value remains?

A.-2 ✅

B.2

C.-8

D.8

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Five zero pairs add five positive and five negative counters while keeping the value at $-3$ . Removing the five negative counters leaves five positive and three negative counters, which simplify to $2$ . Thus $-3-(-5)=2$ .

Q12. A number-line interpretation of a counter model starts at $-2$ and shows subtraction of $-4$ . Which movement and final value are consistent with the counter model?

A.Move

4

units left to

-6

B.Move

4

units right to

2

✅

C.Move

2

units right to

0

D.Move

4

units left to

2

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The starting value is $-2$ . Subtracting a negative integer is modeled by removing four negative counters, which produces the same effect as adding four positive counters. Therefore, the value moves four units right to $2$ .

Q13. Two students model $5-(-3)$ . Student A adds three zero pairs and removes three negative counters. Student B simply removes three positive counters from the original five. Which comparison is correct?

A.Both are correct and give

2

B.Student A is correct and gives

8

; Student B incorrectly gives

2

✅

C.Student B is correct and gives

8

; Student A incorrectly gives

2

D.Neither is correct because subtraction cannot be modeled with counters

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Student A correctly represents subtracting negative three by removing three negative counters. Adding zero pairs makes those negative counters available, leaving eight positive counters. Student B removes positive counters, which models subtracting $3$ , not subtracting $-3$ .

Q14. A model represents $-1$ . You must model subtracting $4$ , but there is only one negative counter and no positive counters. What is the minimum number of zero pairs that must be added so that the subtraction can be completed, and what is the final value?

A.3 zero pairs;

-5

✅

B.4 zero pairs;

-5

C.3 zero pairs;

3

D.4 zero pairs;

5

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: To subtract $4$ , four positive counters must be available. Starting with one negative counter, adding three zero pairs creates three positive and four negative counters. Removing four positive counters leaves four negative counters, giving $-5$ .

Q15. Which expression is equivalent to $12-(-7)$ when subtraction is rewritten as addition?

12+7

✅

12-7

-12+7

-12-7

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The subtraction property changes subtraction into addition of the opposite. Since the opposite of $-7$ is $7$ , $12-(-7)$ becomes $12+7$ , which equals $19$ .

Q16. A student rewrites $-8-5$ as $-8+5$ . Which error has occurred, and what is the correct equivalent expression?

A.The student changed the first number; correct form is

8+(-5)

B.The student used

5

instead of

-5

; correct form is

-8+(-5)

✅

C.The student should change both signs; correct form is

8+5

D.There is no error because subtraction always changes to addition of the same number

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: When subtraction is rewritten as addition, only the number being subtracted changes to its opposite. Thus $-8-5=-8+(-5)$ . The student incorrectly changed positive $5$ into negative $5$ without preserving the subtraction structure correctly.

Q17. A submarine is $18$ meters below sea level and descends another $7$ meters. Which expression correctly models its new position using addition rather than subtraction?

-18+(-7)=-25

✅

-18-(-7)=-11

18+(-7)=11

-18+7=-11

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The submarine begins at $-18$ meters and moves downward $7$ meters, represented by adding $-7$ . Therefore, $-18+(-7)=-25$ . The subtraction property gives the same result if the situation is expressed appropriately.

Q18. Consider $9-13$ . One student calculates $9+(-13)$ , while another calculates $-9+13$ . Which comparison is correct?

A.Both are equivalent because the signs are reversed

B.Only the first is equivalent; both expressions evaluate to

-4

✅

C.Only the second is equivalent because subtraction changes the first number

D.Neither is equivalent because subtraction cannot be rewritten as addition

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The subtraction property changes only the second number to its opposite, so $9-13=9+(-13)$ . The alternative $-9+13$ changes both signs and therefore represents $4$ , not $-4$ .

Q19. A number-line graph shows a starting point at $6$ and a movement $9$ units to the left, ending at $-3$ . Which subtraction expression and equivalent addition expression describe this movement?

6-9=6+(-9)

✅

6-(-9)=6+9

-3-6=-3+6

9-6=9+(-6)

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Starting at $6$ and moving $9$ units left means subtracting $9$ . Using the subtraction property, $6-9$ becomes $6+(-9)$ , which equals $-3$ . This matches the graph's starting and ending positions.

Q20. A store records a balance of $-\15\). A charge of $-\8$ is removed from the recorded balance. Which expression correctly represents the resulting balance, and what is its value?

-15-(-8)=-7

✅

-15+(-8)=-23

-15-8=-23

15+8=23

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Removing a negative charge is represented by subtracting $-8$ . Applying the subtraction property gives $-15+8=-7$ . The key reasoning is that the opposite of the subtracted value $-8$ is positive $8$ .

Q21. For integers $a$ and $b$ , a student claims that $a-b$ and $a+(-b)$ are equivalent only when $b$ is positive. Which example most strongly disproves the claim?

10-3=10+(-3)

4-(-6)=4+6

✅

-7-2=-7+(-2)

8-5=8+(-5)

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The equivalence holds for every integer $b$ , including negative values. With $b=-6$ , $4-(-6)=4+6=10$ . This example directly disproves the claim that the property requires $b$ to be positive.

Q22. Which expression gives the same value as $14-(-9)$ , and why?

14-9

, because the negative sign disappears

14+9

, because subtracting a negative adds its opposite ✅

-14+9

, because both signs must change

-14-9

, because subtraction reverses every sign

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Subtracting a negative number means adding its positive opposite. Therefore, $14-(-9)=14+9=23$ . Only the second option correctly preserves the original first number while changing the subtracted negative into a positive.

Q23. A student evaluates $-11-(-6)$ as $-17$ . Which explanation best identifies the error and corrects the calculation?

A.The student should change

-11

11

, giving

17

B.The student should subtract

6

from

-11

, giving

-17

C.The student treated subtracting

-6

as subtracting

6

; the correct result is

-5

✅

D.The student should add both negative numbers, giving

-5

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The expression $-11-(-6)$ becomes $-11+6$ because the negative being subtracted changes to its positive opposite. This gives $-5$ . The student's mistake was treating subtraction of a negative as subtraction of a positive.

Q24. A bank account has a balance of $-\24\). A negative adjustment of $-\9$ is removed from the account record. What is the resulting balance?

-\$33

-\$15

✅

\$15

\$33

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Removing a negative adjustment is modeled by $-24-(-9)$ . Subtracting the negative is equivalent to adding $9$ , so $-24+9=-15$ . The balance becomes less negative because a negative amount was removed.

Q25. A number-line graph starts at $-7$ and ends at $4$ . The movement is described as subtracting a negative integer. Which expression correctly represents the movement?

-7-11=-18

-7-(-11)=4

✅

-7+(-11)=4

7-(-11)=18

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The point moves from $-7$ to $4$ , a change of $11$ units to the right. Moving right by $11$ can be represented as adding $11$ , or equivalently subtracting $-11$ : $-7-(-11)=4$ .

Q26. Two students simplify $5-(-12)+(-4)$ . Student A changes $5-(-12)$ to $5+12$ , while Student B changes it to $5-12$ . Which student is correct, and what is the final value?

A.Student A;

13

✅

B.Student A;

21

C.Student B;

-11

D.Student B;

13

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Student A correctly changes subtraction of $-12$ into addition of $12$ : $5+12-4=17-4=13$ . Student B incorrectly treats the negative number as though it were positive, producing the wrong result.

Q27. A temperature changes from $-3^\circ\text{C}$ to $8^\circ\text{C}$ after a decrease of a negative amount. Which expression best models the situation?

-3-8=-11

-3-(-11)=8

✅

-3+(-11)=8

3-(-11)=14

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The temperature changes by $11^\circ\text{C}$ upward, from $-3$ to $8$ . Describing this as subtracting a negative amount gives $-3-(-11)$ . Since subtracting $-11$ is adding $11$ , the result is $8$ .

Q28. For integers $x$ and $y$ , suppose $x-y$ is greater than $x$ . What must be true about $y$ , and why?

y

must be positive because subtraction increases the value

y

must be zero because zero does not change

x

y

must be negative because subtracting a negative adds a positive ✅

y

must equal

x

because the result is greater

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: If $x-y>x$ , subtracting $y$ must increase the starting value. This occurs when $y$ is negative, because $x-y=x+(-y)$ , and $-y$ is then positive. Thus the result becomes larger than $x$ .

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