|7| = 7
7 is 7 units from zero.
Free Pre-Algebra Guide
Absolute Value
Learn absolute value as distance from zero, evaluate absolute value expressions, compare distances, understand opposite numbers, and solve absolute value equations step by step.
Absolute value tells us how far a number is from zero on the number line. Because distance cannot be negative, absolute value is always zero or positive.
This distance interpretation explains why opposite numbers such as −6 and 6 have the same absolute value, why |−8| = 8, and why an equation such as |x| = 5 usually has two solutions.
On This Page
- What is absolute value?
- Absolute value as distance from zero
- Opposite numbers and absolute value
- How to evaluate absolute value
- Expressions inside absolute value bars
- Comparing absolute values
- Solving |x| = a
- One solution and no solution
- Shifted absolute value equations
- Worked examples
- Common mistakes
- Reliable strategy
- Interactive practice
What Is Absolute Value?
The absolute value of a number is its distance from zero.
|x| = distance from x to 0
The vertical bars are called absolute value bars. They do not mean multiplication or parentheses.
|−7| = 7
−7 is also 7 units from zero.
Absolute value is about distance, not about
changing a negative sign.
The rule “make it positive” sometimes gives the correct answer, but it hides the actual mathematical meaning.
Absolute Value as Distance From Zero
Consider the number −6. On a number line, −6 lies six units to the left of zero.
−8
−6
−4
−2
0
2
4
6
−6
6 units
|−6| = 6
Distance is never negative.
A point may lie to the left of zero, but the distance from that point to zero is still positive.
Opposite Numbers Have the Same Absolute Value
Opposite numbers lie on opposite sides of zero but are the same distance from zero.
−6
0
6
−6
6
|−6| = |6| = 6
This relationship is the foundation for solving many absolute value equations.
How to Evaluate Absolute Value
To evaluate an absolute value, determine the distance of the number inside the bars from zero.
1. Identify the number
Look at the value inside the absolute
value bars.
2. Think distance
Determine how many units the number lies
from zero.
3. State the distance
Write the nonnegative distance as the
absolute value.
|−9| = 9
|4| = 4
|0| = 0
Expressions Inside Absolute Value Bars
Absolute value bars also act as grouping symbols. If there is arithmetic inside the bars, simplify that expression first.
|−4 + 9|
1. Simplify inside
−4 + 9 = 5
2. Rewrite
|−4 + 9| = |5|
3. Evaluate
|5| = 5
Therefore,
|−4 + 9| = 5.
Do not take the absolute value of each term
separately.
For example, |−4 + 9| is not the same as |−4| + |9|.
Comparing Absolute Values
When comparing absolute values, first evaluate each absolute value and then compare the results.
|−7| ___ |−5|
1. Evaluate the first
|−7| = 7
2. Evaluate the second
|−5| = 5
3. Compare
7 > 5
|−7| > |−5|
Compare absolute values, not the original signed numbers.
Even though −7 is less than −5, its absolute value is greater because −7 is farther from zero.
How to Solve |x| = a
The equation |x| = 6 asks: Which numbers are exactly six units from zero?
|x| = 6
x = −6
−6 is six units from zero.
x = 6
6 is also six units from zero.
The solutions are
x = −6 or x = 6.
Positive distance usually gives two solutions.
For |x| = a where a > 0, the two solutions are x = −a and x = a.
Absolute Value Equations With One or No Solutions
|x| = 0
Only one number is zero units from zero: zero itself.
x = 0
|x| = −4
Absolute value represents distance, and distance cannot be negative.
No solution
| Equation | Number of Solutions | Reason |
|---|---|---|
| |x| = 5 | 2 | Two numbers are 5 units from zero |
| |x| = 0 | 1 | Only zero is 0 units from zero |
| |x| = −5 | 0 | Distance cannot be negative |
Shifted Absolute Value Equations
In an equation such as |x − 3| = 4, the center is no longer zero.
|x − 3| = 4
This means:
x is 4 units from 3.
1. Find the center
Center = 3
2. Move left
3 − 4 = −1
3. Move right
3 + 4 = 7
The solutions are
x = −1 or x = 7.
Worked Absolute Value Examples
Evaluate |−12|
- −12 is 12 units from zero.
- Distance is nonnegative.
- |−12| = 12
Evaluate |3 − 10|
- 3 − 10 = −7.
- Rewrite as |−7|.
- |−7| = 7
Compare |−9| and |4|
- |−9| = 9.
- |4| = 4.
- 9 > 4.
- |−9| > |4|
Solve |x| = 8
- x must be 8 units from zero.
- One point is −8.
- The other point is 8.
- x = −8 or x = 8
Common Absolute Value Mistakes
Mistake 1: Saying |−7| = −7
Absolute value represents distance. The distance from −7 to zero is 7.
Mistake 2: Thinking absolute value means “change the sign”
Positive numbers keep the same value: |5| = 5. The real rule is distance from zero.
Mistake 3: Forgetting to simplify inside the bars
In |−3 + 8|, calculate −3 + 8 first, then evaluate |5|.
Mistake 4: Comparing the original signed numbers
For |−9| and |−4|, compare 9 and 4, not −9 and −4.
Mistake 5: Giving only one solution to |x| = 6
Both −6 and 6 are six units from zero.
Mistake 6: Trying to solve |x| = −5
Absolute value cannot be negative, so this equation has no real solution.
A Reliable Absolute Value Strategy
1. Look inside the bars
If there is arithmetic inside, simplify it first.
2. Think distance
Ask how far the resulting number is from zero.
3. Keep distance nonnegative
Absolute value cannot be negative.
4. For comparisons, evaluate first
Compare the resulting distances.
5. For equations, think both directions
A positive distance usually creates
one point to the left and one to the right.
6. Check special cases
Distance 0 gives one solution; negative
distance gives no solution.
Quick memory aid:
Absolute value = distance.
Practice What You Learned
Try the Absolute Value Practice Tool
Practice evaluating absolute values, interpreting distance from zero, comparing absolute values, finding missing values, and solving absolute value equations with an interactive number line.
Start Absolute Value PracticeRelated Math Resources
Absolute value connects naturally to negative numbers, the coordinate plane, inequalities, and distance.
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