📝 Simplify Expressions with Integers (15 MCQs)

📖 From Digital SAT Algebra • 1. Basics of Algebra • 15 questions available

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What is Simplify Expressions with Integers?

Definition:
Simplifying expressions with integers means reducing an algebraic expression to its simplest form by performing all integer operations (addition, subtraction, multiplication, division) correctly, following the order of operations, and combining like terms to produce a cleaner, equivalent expression.

Working:
Apply PEMDAS: handle parentheses, then exponents (if any), then multiplication/division from left to right, then addition/subtraction from left to right, while always applying integer rules for signs; combine any like variable terms and constant terms at the end.

Example:
Simplify $2 \times (-3) + 4 - 6 \div 2$ .
Solution: $2 \times (-3) = -6$ , $6 \div 2 = 3$ , so expression becomes $-6 + 4 - 3 = -5$ .

Reason:
Simplification makes expressions easier to evaluate, compare, and use in equations, and it ensures that computations are error-free, which is essential for solving complex problems in algebra and higher mathematics.

Easy

Medium

Hard

📝 All Simplify Expressions with Integers MCQs

Q1. Which expression has the same value as $8-(-5)+(-3)$ after correctly simplifying the signs?

A.-10 ✅

B.-6

C.-16

D.0

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: First, subtracting a negative means adding its opposite, so $8-(-5)=13$ . Then add $-3$ , giving $13+(-3)=10$ . The distractors reflect common mistakes such as treating subtraction of a negative as ordinary subtraction.

Q2. A student simplifies $-7+4-(-6)$ as $-7+4-6=-9$ . Which explanation best identifies the student's error?

A.The student should add

7

and

4

first.

B.The student changed subtraction of a negative into subtraction of a positive instead of addition. ✅

C.The student should multiply all terms before adding.

D.The student incorrectly changed

-7

into

7

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The expression contains $-(-6)$ , which is equivalent to $+6$ . The correct simplification is $-7+4+6=3$ . The student's error comes specifically from losing the double-negative relationship.

Q3. A bank account changes by $-\<span class="katex-error" title="ParseError: KaTeX parse error: Can't use function '$ ' in math mode at position 3: 18\̲)̲, then $+\" style="color:#cc0000">18$ , then $+\</span>25$ , and then $-\$12$ . Which simplified integer expression correctly represents the final change, and what is its value?

-18+25-12=-5

✅

18-25+12=5

-18-25-12=-55

18+25-12=31

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The changes must retain their signs: $-18+25+(-12)$ . Combining the first two changes gives $7$ , and subtracting $12$ gives $-5$ . Thus the account's net change is a decrease of $5.

Q4. A student claims that $-4-(-9)+(-6)$ equals $-19$ . Another student says it equals $-1$ . Which conclusion is correct, and why?

A.The first student is correct because both negative numbers must be subtracted.

B.The second student is correct because

-4+9-6=-1

. ✅

C.Both are incorrect because subtraction cannot be used with negative integers.

D.The first student is correct because

-(-9)=-9

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The negative sign before $-9$ changes the operation to addition: $-4-(-9)=-4+9=5$ . Then $5+(-6)=-1$ . The first student's reasoning incorrectly treats subtraction of a negative as subtraction of its magnitude.

Q5. On a number line, a point starts at $-6$ , moves $9$ units to the right, and then moves $5$ units to the left. Which integer expression models the movement and where does the point finish?

-6+9-5=-2

✅

-6-9+5=-10

6+9-5=10

-6+9+5=8

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Moving right represents adding a positive integer, while moving left represents adding a negative integer. Therefore the model is $-6+9+(-5)$ , which simplifies to $3-5=-2$ . The point finishes at $-2$ .

Q6. Two methods are proposed for simplifying $-12-(-7)+(-4)$ . Method A changes subtraction of $-7$ into addition of $7$ . Method B changes it into subtraction of $7$ . Which comparison is correct?

A.Only Method B is valid, giving

-23

B.Both methods are valid because subtracting any number gives the same result.

C.Only Method A is valid, giving

-9

. ✅

D.Neither method is valid because negative numbers cannot be simplified.

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Method A correctly rewrites $-12-(-7)+(-4)$ as $-12+7-4$ , producing $-9$ . Method B incorrectly treats subtraction of a negative as subtraction of a positive, which changes the value and produces an invalid result.

Q7. A graph represents four successive integer changes: starting at $3$ , moving $7$ units left, $11$ units right, and $6$ units left. Which expression and final position correctly describe the graph?

3-7+11-6=1

✅

3+7-11+6=5

3-7-11-6=-21

3+7+11-6=15

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: A leftward movement is represented by subtraction and a rightward movement by addition. Starting from $3$ , the sequence is $3-7+11-6$ . This gives $-4+11-6=7-6=1$ , so the final position is $1$ .

Q8. Which value of $x$ makes the expression $3x-(-8)+(-5)$ equal to $12$ ?

x=1

x=2

✅

x=3

x=5

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Rewrite the expression as $3x+8-5=3x+3$ . Setting this equal to $12$ gives $3x+3=12$ , so $3x=9$ and $x=3$ . Therefore option C is correct; the other choices result from sign or arithmetic errors.

Q9. What is the value of $18-3\times(-4)+(-6)\div2$ ?

A.-27

B.-21

C.-33 ✅

D.-15

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: Multiplication and division must be completed before addition and subtraction. Thus $3\times(-4)=-12$ and $(-6)\div2=-3$ . The expression becomes $18-(-12)-3=18+12-3=27$ , so the correct answer is A.

Q10. A student evaluates $-5+2\times[7-(-3)]$ as $17$ . Which step caused the error?

A.The student added

-5

and

2

first.

B.The student treated

-3

as positive inside the brackets. ✅

C.The student multiplied

2

7

before simplifying the brackets.

D.The student subtracted

5

from

2

after multiplication.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: First simplify the brackets: $7-(-3)=10$ . Then multiply $2\times10=20$ , and finally add $-5$ , giving $15$ . The student's result of $17$ shows a sign error involving the subtraction of a negative integer.

Q11. A temperature starts at $-8^\circ\text{C}$ , rises by $3\times4^\circ\text{C}$ , and then decreases by $10\div2^\circ\text{C}$ . Which expression models the situation correctly?

-8+3\times4-10\div2

✅

-8+3)\times4-10\div2

-8+3\times(4-10)\div2

-8-3\times4+10\div2

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The initial temperature is $-8$ . A rise of $3\times4$ means adding $12$ , while a decrease of $10\div2$ means subtracting $5$ . The correct model is $-8+3\times4-10\div2$ , which gives $-1^\circ\text{C}$ .

Q12. Which comparison correctly evaluates $24\div(-6)+3\times(-2)-(-5)$ ?

A.Multiplication first gives

-4+(-6)-(-5)=-5

B.Division and multiplication first give

-4-6+5=-5

. ✅

C.Subtraction of the negative must happen first, giving

24\div(-1)+3\times(-2)=-30

D.Adding all negative terms first gives

-15

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Division and multiplication have priority and are evaluated before addition or subtraction. The expression becomes $-4+(-6)-(-5)$ . Since subtracting $-5$ means adding $5$ , the result is $-4-6+5=-5$ .

Q13. A number-line model starts at $-2$ . One operation represents moving $3$ groups of $4$ units left, followed by moving $5$ units right. Which final position matches the model?

A.-15

B.-9 ✅

C.-5

D.-1

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Three groups of four units left represent $-3\times4=-12$ . Starting at $-2$ gives $-2+(-12)=-14$ . Moving $5$ units right adds $5$ , resulting in $-9$ . The key is interpreting multiplication as repeated movement before combining changes.

Q14. Two students simplify $-18\div3-2[4-(-1)]$ . Student A gets $-16$ , while Student B gets $-4$ . Which statement is correct?

A.Student A is correct because subtraction is performed before multiplication. ✅

B.Student B is correct because the bracket must be simplified before multiplication.

C.Both students are correct because the operations can be performed in any order.

D.Neither is correct; the expression equals

4

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: First simplify the bracket: $4-(-1)=5$ . Then calculate $-18\div3=-6$ and $2\times5=10$ . Finally, $-6-10=-16$ . Student A correctly followed the required operation sequence, while Student B incorrectly combined terms before completing the multiplication.

Q15. Which integer $x$ satisfies $20-2[3x-(-4)]=6$ ?

x=2

x=3

✅

x=4

x=5

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Simplify the bracket first: $3x-(-4)=3x+4$ . Then $20-2(3x+4)=6$ , so $20-6x-8=6$ . This gives $12-6x=6$ , hence $-6x=-6$ and $x=1$ , so none of the listed options is correct.

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