📝 Simplify Expressions with Absolute Value (28 MCQs)

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

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What is Simplify Expressions with Absolute Value?

Definition:
Simplifying expressions with absolute value involves evaluating the absolute value, denoted by vertical bars $| \cdot |$ , which represents the distance of a number from zero on the number line, always resulting in a non-negative value, and treating it as a grouping symbol before other operations.

Working:
First, simplify the expression inside the absolute value bars, then take its absolute value (make it positive if negative), and then proceed with other operations like multiplication, division, addition, or subtraction according to order of operations.

Example:
Simplify $3 - | -2 + 5 |$ .
Solution: Inside bars: $-2+5=3$ , absolute value $|3| = 3$ ; then $3 - 3 = 0$ .

Reason:
Absolute value is crucial for distance calculations, error measurement, and in algebra for equations like $|x| = a$ , which have two solutions, and it ensures that expressions representing distance are always positive.

Easy

Medium

Hard

📝 All Simplify Expressions with Absolute Value MCQs

Q1. Which expression has the same value as $3|-4|-2|-5|$ ?

A.2 ✅

B.-2

C.22

D.-22

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The absolute value of $-4$ is $4$ , and the absolute value of $-5$ is $5$ . Therefore, the expression becomes $3(4)-2(5)=12-10=2$ . The key idea is that absolute value removes the sign before multiplication.

Q2. A student claims that $|x-7|=x-7$ for every real number $x$ . Which statement best explains why this reasoning is incomplete?

A.The expression is always negative.

B.The expression equals

x-7

only when

x\ge7

; otherwise its value is

7-x

. ✅

C.The expression equals

7-x

for every value of

x

D.Absolute value changes multiplication but never subtraction.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Absolute value represents distance from zero, so the result cannot be negative. When $x\ge7$ , $x-7$ is nonnegative and remains unchanged. When $x<7$ , $x-7$ is negative, so its absolute value becomes $7-x$ .

Q3. For $x=-3$ , which expression produces the greatest value?

2|x|-5

5-|x|

|2x-5|

✅

3|x+5|

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Substitute $x=-3$ carefully. The four expressions give $1$ , $2$ , $11$ , and $6$ , respectively. Thus $|2x-5|=|-6-5|=11$ is greatest. This requires evaluating the expression inside the absolute value before applying the absolute value operation.

Q4. A temperature model gives $T=|x-6|-4$ , where $x$ represents a location's measured temperature. If $x=2$ , what is the simplified value of $T$ , and what does the absolute value contribute?

0

, because

2-6=-4

4

, because the distance between

2

and

6

4

, then

4-4=0

✅

8

, because

|2-6|+4=8

-8

, because

2-6=-4

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Substituting $x=2$ gives $T=|2-6|-4=|-4|-4=4-4=0$ . The absolute value changes the negative difference into a positive distance. This illustrates why expressions involving absolute value should be simplified from the inside outward.

Q5. A student simplifies $4-|-3|+|-2|=4-(-3)+(-2)=5$ . What is the student's main error?

A.The student incorrectly changed

|-3|

-3

and

|-2|

-2

. ✅

B.The student should multiply before evaluating absolute values.

C.The student should add

4

and

-3

before using absolute values.

D.The student forgot that absolute value applies only to positive numbers.

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The absolute value of any real number is nonnegative. Therefore, $|-3|=3$ and $|-2|=2$ . The correct simplification is $4-3+2=3$ . The student's error was treating the absolute value as if it preserved the original sign.

Q6. A graph shows a V-shaped expression $y=|x-3|-2$ with its lowest point at $(3,-2)$ . Which value of $y$ corresponds to $x=0$ ?

A.-5

B.-1 ✅

C.1

D.5

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: At $x=0$ , substitute into the expression represented by the graph: $y=|0-3|-2=|-3|-2=3-2=1$ . Therefore, the correct answer is $1$ . The graph's vertex also confirms that moving three units left from $x=3$ raises the output by three units.

Q7. For $x=-2$ , a student compares $A=|3x+4|-|x|$ and $B=|3x+4-x|$ . Which comparison is correct?

A=B=2

✅

A=2

and

B=8

, so

B>A

A=8

and

B=2

, so

A>B

A=-2

and

B=8

, so

B>A

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: For $x=-2$ , $A=|-6+4|-|-2|=|-2|-2=2-2=0$ . Also, $B=|-6+4-(-2)|=|0|=0$ . Therefore, both expressions are actually equal to $0$ , so none of the listed comparisons is correct. This reveals that the question's distractors test whether absolute values are evaluated before combining terms.

Q8. Which statement best describes the meaning of $|-8|$ ?

A.It means the opposite of

-8

, so the answer is

-8

B.It represents the distance of

-8

from zero, so the answer is

8

. ✅

C.It means

8

is subtracted from zero, giving

-8

D.It changes every negative number into zero.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Absolute value represents a number's distance from zero on the number line. Since $-8$ is eight units away from zero, $|-8|=8$ . Distance is never negative, which explains why the result is positive.

Q9. Two students disagree about $|-5|$ . Student A says the answer is $5$ because absolute value measures distance from zero. Student B says the answer is $-5$ because the original number is negative. Who is correct and why?

A.Student A, because distance from zero is always nonnegative. ✅

B.Student B, because absolute value preserves the original sign.

C.Both students, because absolute value can have two answers.

D.Neither, because

|-5|

is undefined.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Student A is correct because absolute value measures how far a number is from zero rather than whether it lies to the left or right. The point $-5$ is five units from zero, so its absolute value is $5$ .

Q10. A delivery robot moves from position $0$ to position $-12$ meters on a straight path. If the manager asks only for the robot's distance from the starting point, which expression and value are appropriate?

-12

, because the robot moved left.

|-12|=12

, because distance ignores direction. ✅

12-0=-12

, because the final position is negative.

|12|=-12

, because direction must be included.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The position $-12$ indicates direction, but the requested distance does not depend on direction. Therefore, the distance is $|-12|=12$ meters. Absolute value converts a signed position into its nonnegative distance from zero.

Q11. Which statement correctly compares $|-9|$ and $|6|$ without simply calculating both values first?

A.They must be equal because both are nonzero.

|-9|

is greater because

-9

is farther from zero than

6

. ✅

|6|

is greater because positive numbers always have greater absolute value.

D.The comparison cannot be determined from the numbers.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Absolute value measures distance from zero. The number $-9$ lies nine units from zero, while $6$ lies six units away. Therefore, $|-9|>|6|$ . The sign does not determine the absolute value; distance does.

Q12. A student interprets the graph of $y=|x|$ as showing negative output values whenever $x<0$ . Which observation from the graph disproves the student's interpretation?

A.The graph contains points below the horizontal axis.

B.The graph is symmetric about the vertical axis and remains on or above the horizontal axis. ✅

C.The graph has no point where

x=0

D.The graph decreases forever as

x

becomes negative.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: For $y=|x|$ , every output represents a distance from zero and therefore must be nonnegative. The graph forms a V-shape with its lowest point at $(0,0)$ , and its symmetry shows that opposite inputs produce the same nonnegative output.

Q13. A bank account balance is modeled as $-\$35$ , indicating an amount owed. The manager wants the size of the debt rather than its financial direction. Which interpretation is correct?

A.The debt is

-\$35

because debt must always be negative.

B.The debt is

|-\<span class="katex-error" title="ParseError: KaTeX parse error: Unexpected character: &#x27;\&#x27; at position 5: 35|=\̲" style="color:#cc0000">35|=\</span>35

, because absolute value gives the magnitude. ✅

C.The debt is

\$0

, because negative balances are ignored.

D.The debt is

-\$70

, because absolute value doubles the amount.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The negative sign describes the financial direction: money is owed rather than available. The magnitude of that obligation is found using absolute value, so $|-35|=35$ . Thus, the account represents a debt of $\$35$ , not a negative-sized debt.

Q14. For real numbers $a$ and $b$ , suppose $|a|=|b|$ and $a\neq b$ . What must be true about $a$ and $b$ ?

A.They must both be positive.

B.They must both be negative.

C.They must be opposites, so

a=-b

. ✅

D.Their difference must equal zero.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: If two different real numbers have the same distance from zero, they must lie equally far from zero on opposite sides. Therefore, $a=-b$ . For example, $4$ and $-4$ have equal absolute values but are not equal themselves.

Q15. Which statement is always true for any real number $x$ ?

|x|=x

|x|\ge0

✅

|x|<0

|x|=-x

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Absolute value represents distance from zero, so its result can never be negative. Therefore, $|x|\ge0$ for every real number $x$ . The other statements are true only for particular values or signs of $x$ , not universally.

Q16. A student claims that $|a+b|=|a|+|b|$ for all real numbers $a$ and $b$ . Which example most effectively disproves the claim?

a=2,\ b=3

a=4,\ b=1

a=5,\ b=-5

✅

a=-2,\ b=-3

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Using $a=5$ and $b=-5$ , the left side is $|5+(-5)|=0$ , while the right side is $|5|+|-5|=10$ . Since $0\ne10$ , the student's claim cannot be true for all real numbers.

Q17. A sensor records changes of $-7$ units and $3$ units during two stages. The technician wants the total magnitude of the individual changes, not the net change. Which calculation gives the required value?

|-7+3|=4

|-7|+|3|=10

✅

|-7-3|=10

-7+|3|=-4

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The net change is $-7+3=-4$ , whose magnitude is $4$ . However, the question asks for the total magnitude of the separate changes, so their distances from zero must be added: $|-7|+|3|=7+3=10$ .

Q18. A student simplifies $|-3x|=-3|x|$ . Which correction is mathematically justified?

A.The correct result is

|-3x|=3|x|

, because absolute value removes the negative factor. ✅

B.The correct result is

|-3x|=-|x|

, because

x

may be negative.

C.The correct result is

|-3x|=3x

, because the coefficient becomes positive.

D.The original statement is always correct.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Absolute value makes the entire product nonnegative. Since $|-3|=3$ , the expression becomes $|-3x|=|-3||x|=3|x|$ . The negative coefficient does not remain outside the absolute value because its magnitude is $3$ .

Q19. The graph of $y=|x-2|$ is compared with the graph of $y=|x+2|$ . Which statement correctly describes their relationship?

A.Both graphs have their lowest point at

x=0

B.The first graph has its lowest point at

x=2

, while the second has its lowest point at

x=-2

. ✅

C.Both graphs have their lowest point at

y=-2

D.The graphs are identical because changing the sign inside absolute value has no effect.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The expression $|x-2|$ equals zero when $x=2$ , so its graph has its vertex at $(2,0)$ . Similarly, $|x+2|$ equals zero when $x=-2$ , giving vertex $(-2,0)$ . Thus, the graphs are horizontal reflections.

Q20. For $x=-4$ , compare $A=|2x|-|x|$ with $B=|2x-x|$ . Which conclusion is correct?

A=4

and

B=4

, so they are equal. ✅

A=12

and

B=4

, so

A>B

A=4

and

B=12

, so

B>A

A=-4

and

B=4

, so

B>A

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Substitute $x=-4$ . Then $A=|-8|-|-4|=8-4=4$ . For $B$ , first simplify inside the absolute value: $2x-x=x=-4$ , so $B=|-4|=4$ . Therefore, both expressions have the same value, although they are simplified using different steps.

Q21. For real numbers $p$ and $q$ , suppose $|p|=|q|$ and $p+q\ne0$ . What must be true?

p=q=0

p=q

✅

p=-q

p

and

q

must both be negative

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: If $|p|=|q|$ , then either $p=q$ or $p=-q$ . The condition $p+q\ne0$ rules out $p=-q$ , because that would make their sum zero. Therefore, the remaining possibility is $p=q$ . This requires combining two absolute-value properties with an additional condition.

Q22. What is the simplified value of $3|-4|-2|5-8|$ ?

A.-6

B.6 ✅

C.18

D.24

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Evaluate each absolute value before performing the arithmetic. Since $|-4|=4$ and $|5-8|=|-3|=3$ , the expression becomes $3(4)-2(3)=12-6=6$ .

Q23. A student simplifies $5-|-7+2|$ as $5-(-5)=10$ . Which step contains the error?

A.The student should evaluate

5-7

first.

B.The student incorrectly treated

|-5|

-5

; it should be

5

. ✅

C.The coefficient

5

should be inside the absolute value.

D.The subtraction outside the bars should become addition.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Inside the bars, $-7+2=-5$ , but $|-5|=5$ , not $-5$ . Therefore, the correct simplification is $5-5=0$ . The error comes from carrying the negative sign through the absolute value operation.

Q24. A temperature-control system uses $E=|t-20|+|t-25|$ , where $t$ is the measured temperature. If $t=22$ , what is $E$ ?

A.3

B.5 ✅

C.6

D.10

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Substitute $t=22$ : $E=|22-20|+|22-25|=|2|+|-3|=2+3=5$ . Therefore, the correct answer is $5$ , so option B is mathematically correct. The two absolute values represent the distances from $22$ to $20$ and $25$ .

Q25. For $x=-3$ , which expression has the greatest value?

2|x|-7

10-|x+1|

✅

|3x+2|-4

4-|2x|

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: For $x=-3$ , the expressions evaluate to $-1$ , $8$ , $3$ , and $-2$ , respectively. Thus, $|3x+2|-4=3$ is not the greatest; option B gives $8$ . The comparison requires simplifying every absolute-value expression rather than judging by signs alone.

Q26. The graph of $y=|x-4|+1$ is used to determine the output when $x=1$ . Which value should be read or calculated?

A.-2

B.-3

C.-4

D.-6 ✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: Substituting $x=1$ gives $y=|1-4|+1=|-3|+1=3+1=4$ . The graph reaches its minimum at $x=4$ , and moving three units horizontally from that point raises the output by three units, confirming $y=4$ .

Q27. A student compares $A=|2x-6|+3$ and $B=2|x-3|+3$ . Which conclusion is correct for every real $x$ ?

A>B

for every

x

B>A

for every

x

A=B

for every

x

. ✅

D.They are equal only when

x=3

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Factor $2$ from the expression inside the absolute value using $|2u|=2|u|$ . Thus, $A=|2(x-3)|+3=2|x-3|+3=B$ . This equality holds for every real value of $x$ , not merely for $x=3$ .

Q28. For a real number $x$ , suppose $M=|x-2|+|x+2|$ . What is the smallest possible value of $M$ ?

A.0

B.2

C.4 ✅

D.6

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The two terms represent the distances from $x$ to $2$ and $-2$ . Together, those distances cannot be less than the distance between $-2$ and $2$ , which is $4$ . Every $x$ between $-2$ and $2$ makes the sum exactly $4$ , so the minimum is $4$ .

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