Absolute Value Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Absolute Value.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The distance of a number from zero on the number line, always non-negative; written ∣x∣|x|. For any real number, absolute value strips away the sign and returns only the magnitude.

−5-5 and 55 are both 5 units from zero, so ∣−5∣=∣5∣=5|-5| = |5| = 5.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Absolute value measures how far a number is from zero on the number line.

Common stuck point: The procedure for absolute value is the easy part; the trap is keeping the negative sign inside absolute value. Asking "Is the sign direction important, or only the distance?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the sign direction important, or only the distance?

Worked Examples

Example 1

easy
Evaluate ∣−7∣+∣3∣|{-7}| + |3|.

Answer

1010

First step

1
The absolute value of −7-7 is the distance from 0: ∣−7∣=7|{-7}| = 7.

Full solution

  1. 2
    The absolute value of 33 is ∣3∣=3|3| = 3.
  2. 3
    Add: 7+3=107 + 3 = 10.
Absolute value measures distance from zero on the number line and is always non-negative. For any number aa, ∣a∣=a|a| = a if a≥0a \ge 0 and ∣a∣=−a|a| = -a if a<0a < 0.

Example 2

medium
Evaluate ∣5−12∣−∣2−9∣|5 - 12| - |2 - 9|.

Example 3

medium
Compute ∣∣−3∣−∣−8∣∣\big||{-3}|-|{-8}|\big|.

Example 4

hard
Simplify ∣x−2∣|x-2| when x<2x<2.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Evaluate ∣−15∣−∣−4∣|{-15}| - |{-4}|.

Example 2

medium
Evaluate ∣3−8∣+∣−2∣−∣5∣|3 - 8| + |{-2}| - |5|.

Example 3

easy
Compute ∣−8∣|-8|.

Example 4

easy
Compute ∣7∣|7|.

Example 5

easy
Compute ∣0∣|0|.

Example 6

easy
Compute −∣6∣-|6|.

Example 7

easy
Compute ∣3−8∣|3 - 8|.

Example 8

easy
Compute ∣−2∣+∣5∣|-2| + |5|.

Example 9

easy
What is the distance between −3-3 and 44 on the number line?

Example 10

easy
Solve ∣x∣=5|x| = 5.

Example 11

medium
Solve ∣x−3∣=7|x - 3| = 7.

Example 12

medium
Compute ∣−4∣×∣−3∣|{-4}| \times |{-3}|.

Example 13

medium
Solve ∣2x∣=10|2x| = 10.

Example 14

medium
Compute ∣−2∣3|{-2}|^3.

Example 15

medium
Evaluate ∣x−5∣|x - 5| when x=2x = 2.

Example 16

medium
Solve ∣x∣<4|x| < 4. Write the solution as an interval.

Example 17

medium
Solve ∣x∣>3|x| > 3. Describe the solution.

Example 18

medium
Simplify ∣x∣|x| if you know x<0x < 0.

Example 19

medium
Compute (−6)2\sqrt{(-6)^2}.

Example 20

challenge
Solve ∣2x−1∣=∣x+4∣|2x - 1| = |x + 4|.

Example 21

challenge
Solve ∣x−2∣+∣x+1∣=5|x - 2| + |x + 1| = 5.

Example 22

challenge
What is the minimum value of ∣x−1∣+∣x−5∣|x - 1| + |x - 5|, and where is it achieved?

Example 23

easy
Compute ∣−12∣|-12|.

Example 24

easy
Compute ∣9∣|9|.

Example 25

easy
Compute −∣−5∣-|-5|.

Example 26

easy
Compute ∣10−4∣|10-4|.

Example 27

easy
Compute ∣2−9∣|2-9|.

Example 28

medium
Compute ∣3−7∣+∣7−3∣|3-7|+|7-3|.

Example 29

medium
Compute ∣−4∣⋅∣−3∣+∣2∣|-4|\cdot|-3|+|2|.

Example 30

medium
Evaluate ∣x∣|x| if x=−34x=-\frac{3}{4}.

Example 31

medium
Compute (−9)2\sqrt{(-9)^2}.

Example 32

medium
Order from smallest to largest: ∣−3∣, −∣7∣, ∣−5∣, 0|-3|,\ -|7|,\ |-5|,\ 0.

Example 33

medium
If ∣x∣=8|x|=8, list all possible values of xx.

Example 34

hard
Compute ∣3−π∣+∣π−4∣|3-\pi|+|\pi-4|.

Example 35

hard
For what value of xx is ∣x−3∣+∣x−7∣|x-3|+|x-7| minimized?

Example 36

hard
Find all real xx with ∣x∣+x=0|x|+x=0.

Example 37

medium
Compute ∣2∣3−∣−3∣2|2|^3-|-3|^2.

Example 38

medium
Compute ∣−12∣∣−4∣\frac{|-12|}{|-4|}.

Example 39

challenge
Find the smallest possible value of ∣x−1∣+∣x−4∣+∣x−9∣|x-1|+|x-4|+|x-9| over real xx, and where it occurs.

Example 40

challenge
For what real xx does ∣x−2∣=∣2−x∣|x-2|=|2-x| fail?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

integers