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Absolute Value: Definition, Symbol, Graph & Examples

TL;DR

The absolute value of a number is its distance from zero on the number line, written with vertical bars as ∣x∣, and it is never negative. So ∣5∣=5 and ∣−5∣=5.

What Is Absolute Value?

The absolute value of a number is how far that number sits from zero, regardless of which side it is on. Direction is dropped; only the distance remains.

Because distance cannot be negative, the absolute value of any number is zero or positive.

∣7∣=7 and ∣−7∣=7. Both are 7 units from zero.

The formal definition is a two-case (piecewise) rule:

∣x∣={xif x≥0−xif x<0

The second case looks strange at first. When x is negative, −x is positive. For example, if x=−7, then −x=−(−7)=7. So the rule simply strips the sign and returns the positive size.

The Absolute Value Symbol

Absolute value is written with two vertical bars around the number or expression: ∣x∣, read aloud as "the absolute value of x" or "mod x."

Table 1: Reading the absolute value symbol.

The bars are not parentheses. Everything inside is evaluated first, then the sign is stripped.

Properties of Absolute Value

These rules follow from "distance from zero" and are the ones used most when simplifying.

Table 2: Key properties of absolute value.

The triangle inequality is the one students meet last: the size of a sum is never more than the sum of the sizes, because cancellation can only shrink the total.

The Absolute Value Function And Its Graph

The absolute value function is f(x)=∣x∣. Its graph is a V shape: two straight rays meeting at a single corner point called the vertex.

The general transformed form is:
f(x)=a,∣x−h∣+k
Here (h,k) is the vertex. If a>0 the V opens upward and the vertex is the lowest point; if a<0 it opens downward and the vertex is the highest point. The value of ∣a∣ controls how steep the rays are.

Examples of absolute value

Example 1: Absolute value of a positive number

Find ∣12∣. 12 is already 12 units from zero, on the positive side. Final answer: ∣12∣=12.

Example 2: Absolute value of a negative number

Find ∣−9∣. −9 sits 9 units from zero. Strip the sign: ∣−9∣=−(−9)=9 Final answer: ∣−9∣=9.

Example 3: Absolute value of an expression

Evaluate ∣3−8∣. Work inside the bars first: 3−8=−5. Then take the absolute value: ∣−5∣=5 Final answer: ∣3−8∣=5.

Example 4: Solving a simple absolute value equation

Solve ∣x∣=6. A number is 6 units from zero in two places, to the right and to the left: x=6 or x=−6. Final answer: x=6 or x=−6.

Example 5: A wrong path first, then the fix

Solve ∣x−2∣=4x+1. The intuitive-but-wrong move is to drop the bars and solve one equation: x−2=4x+1, which gives −3=3x, so x=−1. Check it. Substitute x=−1 into the original right side: 4(−1)+1=−3. But the left side ∣x−2∣ can never be negative, so a solution that makes the right side negative is impossible. That makes x=−1 extraneous. The correct way solves two cases and checks both:

Case 1: x−2=4x+1⇒ x = -1 (rejected, as shown above). Case 2: x−2=−(4x+1)⇒ x=15. Check x=15: right side =4(15)+1=95, which is positive, and the left side matches. Final answer: x=15. The other case produced an extraneous solution.

Example 6: An equation with no solution

Solve ∣x∣=−3. Absolute value is a distance, so it is never negative. No number has a negative distance from zero. Final answer: no solution.

Why Absolute Value Exists

Absolute value exists because we often care about size without caring about direction. How far is the store, how big is the error, how much did the temperature change? In each case the sign is noise, and what matters is the magnitude.

The idea shows up across fields:

• Engineering tolerance. A bolt machined to within ∣d−10∣≤0.05 mm is acceptable whether it comes out slightly over or slightly under the target.
• Navigation and GPS. Position error is reported as a distance, never as a signed direction. Being 3 metres off is 3 metres off, north or south.
• Finance. The size of a price swing matters for risk whether the price rose or fell.
• Physics. Speed is the absolute value of velocity; it keeps the magnitude and discards the direction.

The Mathematicians Behind Absolute Value

• Karl Weierstrass (1815–1897, Germany) introduced the now-universal ∣x∣ vertical-bar notation around 1841 while building calculus on rigorous foundations.
• Jean-Robert Argand (1768–1822, France) extended the idea of "size" to complex numbers, where the absolute value becomes the distance from the origin in a two-dimensional plane.

Common Mistakes With Absolute Value

Mistake 1: Dropping the sign too early

Evaluate everything inside the bars first, then strip the sign.

Mistake 2: Believing absolute value can be negative

A distance is never negative, so ∣x∣=−3 has no solution.

Mistake 3: Solving ∣x∣=k as one case instead of two

Set up both cases: x=6 or x=−6. Two points are the same distance from zero.

Mistake 4: Accepting extraneous solutions

Substitute each candidate back into the original equation. Reject any that make the right side negative.

Practice Questions on Absolute Value

Try these:

1. Evaluate ∣−15∣.
2. Evaluate ∣2−9∣.
3. Solve ∣x∣=10.
4. Solve ∣x∣=−4.
5. Solve ∣2x−1∣=7.

Answers

1. ∣−15∣=15.
2. 2−9=−7, so ∣−7∣=7.
3. x=10 or x=−10.
4. No solution — absolute value can never equal a negative number.
5. 2x−1=7⇒x=4, or 2x−1=−7⇒x=−3.

Frequently Asked Questions

Can absolute value be negative? No. Why does ∣x∣=−3 have no solution? Because no number sits a negative distance from zero. What is the absolute value function's graph? A V shape with its vertex at the origin for f(x)=∣x∣, symmetric about the y-axis, opening upward.