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Absolute Value

Absolute Value

Absolute value makes a negative number positive. Positive numbers and 0 are left unchanged. The absolute value of x is written |x|. We write |–6| = 6 and |8| = 8.

Formally, the absolute value of a number is the distance between the number and the origin. This is a much more powerful definition than the "makes a negative number positive" idea. It connects the notion of absolute value to the absolute value of a complex number and the magnitude of a vector.

 

See also

Absolute value rules

Key Formula

∣x∣={xif x≥0−xif x<0|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}
Where:
  • xx = Any real number
  • ∣x∣|x| = The absolute value of x, representing its distance from 0

Worked Example

Problem: Evaluate |3 − 10| + |4|.
Step 1: Simplify inside the first absolute value bars.
3−10=−73 - 10 = -7
Step 2: Apply the absolute value to −7. Since −7 is negative, the absolute value flips it to positive.
∣−7∣=7|-7| = 7
Step 3: Apply the absolute value to 4. Since 4 is already positive, it stays the same.
∣4∣=4|4| = 4
Step 4: Add the two results.
7+4=117 + 4 = 11
Answer: |3 − 10| + |4| = 11

Another Example

Problem: Solve the equation |x − 5| = 3.
Step 1: The equation says the distance between x and 5 is 3. This means x − 5 could be 3 or −3.
x−5=3orx−5=−3x - 5 = 3 \quad \text{or} \quad x - 5 = -3
Step 2: Solve each equation separately.
x=8orx=2x = 8 \quad \text{or} \quad x = 2
Step 3: Check: |8 − 5| = |3| = 3 ✓ and |2 − 5| = |−3| = 3 ✓. Both solutions work.
Answer: x = 8 or x = 2

Frequently Asked Questions

Can absolute value ever be negative?
No. Absolute value measures distance, and distance is never negative. The result of |x| is always zero or positive, regardless of what x is. For example, |−12| = 12 and |0| = 0.
Why does an absolute value equation have two solutions?
Because two different numbers can have the same distance from a point. For instance, |x| = 5 means x is 5 units from 0, which is true for both x = 5 and x = −5. Each absolute value equation splits into two cases: one where the inside is positive and one where it is negative.

Absolute Value vs. Opposite (Negation)

The opposite of a number x is −x, which simply reverses its sign: the opposite of 7 is −7, and the opposite of −3 is 3. Absolute value, on the other hand, always returns a non-negative result. For positive numbers, |x| = x (no sign change), while the opposite would be negative. They only give the same result when x is negative or zero: |−4| = 4 and the opposite of −4 is also 4.

Why It Matters

Absolute value appears whenever you need to express a distance or a magnitude without caring about direction. It is essential in defining error and tolerance — for instance, saying a measurement must be within ∣x−50∣≤2|x - 50| \leq 2 means it can range from 48 to 52. Absolute value also forms the foundation for more advanced ideas like the absolute value of a complex number and the magnitude of a vector.

Common Mistakes

Mistake: Distributing absolute value over addition: writing |a + b| = |a| + |b|.
Correction: Absolute value does not distribute over addition. For example, |−3 + 1| = |−2| = 2, but |−3| + |1| = 3 + 1 = 4. These are not the same. The correct relationship is the triangle inequality: |a + b| ≤ |a| + |b|.
Mistake: Thinking −|x| equals |−x| or that the negative sign outside disappears.
Correction: A negative sign outside the bars stays. −|x| is the negative of the absolute value: −|−6| = −6, not 6. Meanwhile, |−x| = |x|, which is always non-negative.

Related Terms