What does absolute value actually mean?

The distance from zero, with the direction discarded. |−7| and |7| are both 7 because both numbers sit seven units from the origin, one on each side. Formally the definition has two branches: |x| is x when x is at least zero, and −x when x is negative. That minus is not "removing the sign" — it is negating a negative, which is what makes the result come out positive.

Why is √(x²) equal to |x| rather than x?

Because the square root symbol means the non-negative root, and squaring destroys the sign. Put x = −3 in: x² is 9, and √9 is 3, which is |−3| and not −3. Writing √(x²) = x is only correct when x is already non-negative. This is exactly why solving x² = 9 gives two answers: taking the root of both sides produces |x| = 3, which then splits into x = 3 and x = −3.

Why does an absolute value equation have two answers?

Because |a| = |b| means a = b or a = −b, never just the first. When you write |x − 3| = 2 you are saying the inside has size 2, and something can have size 2 by being 2 or by being −2. Both give a valid value of x. Read as a distance it is even clearer: the equation asks which numbers sit 2 away from 3, and there is one on each side.

When does an absolute value equation have fewer than two answers?

When the right-hand side is zero there is exactly one, because the two cases collapse together — only one number is zero distance from anywhere. When the right-hand side is negative there are none at all, since an absolute value is never negative. That second case is worth checking before you start: setting up the two cases anyway produces two numbers that look like answers and satisfy neither.

How do I find the distance between two numbers?

Subtract them and take the absolute value: |a − b|. The order does not matter, because |a − b| and |b − a| are the same — the bars discard the sign that swapping introduces. This is the reading that makes the rest of the subject intuitive: |x| is just |x − 0|, the distance from the origin, and |x − 3| < 2 says x lies within 2 of 3 rather than anything about algebra.

Is |a + b| the same as |a| + |b|?

Not generally, and the difference matters. |3 + (−5)| is 2, while |3| + |−5| is 8 — the numbers partly cancel before the size is taken. The relationship is an inequality: |a + b| is always at most |a| + |b|, and the two are equal exactly when a and b share a sign or one of them is zero. This is the triangle inequality, and it holds for every pair of numbers without exception.

Does absolute value work with multiplication?

Yes, and cleanly — unlike addition. |ab| equals |a| times |b| for every pair, and the same holds for division wherever the denominator is not zero. The reason is that multiplying two numbers multiplies their sizes regardless of their signs, whereas adding them can cancel. So the bars pass straight through a product and can be moved onto each factor freely, which is often the quickest simplification available.

What is the modulus of a complex number?

The same idea in two dimensions. |a + bi| is the distance from the origin to the point (a, b) on the complex plane, which Pythagoras gives as √(a² + b²). So |3 + 4i| is 5. Setting the imaginary part to zero collapses it to the ordinary absolute value — √(a²) is |a|, the identity from before. That is why the same bars are used for both: they mean magnitude in either setting.

What does the graph of an absolute value look like?

A V. The graph of y = |ax + b| turns at the point where the inside changes sign, which is x = −b/a, and that vertex sits on the axis because the expression is zero there. To one side the bars do nothing and the line runs as it would anyway; to the other they reflect it upwards. Nothing ever dips below the axis, which is the definition drawn rather than a separate fact.

Can an absolute value ever be negative?

The value itself never can — it is a distance. But an expression with bars around part of it certainly can: −|x| is negative for every x except zero, and 3 − |x| is negative whenever |x| exceeds 3. The distinction is between what the bars produce, which is never negative, and what happens to it afterwards. Only the first is guaranteed.

Absolute Value Calculator

The absolute value of a number is its distance from zero, and reading it that way makes almost everything else about it obvious. Four questions wear the same pair of bars and are not the same question. Evaluating |ax + b| at a point is arithmetic. Solving |ax + b| = c is an equation with generally ...

√(x²) is |x|, not x. At x = −3 the left side comes to 3. That single fact explains why solving x² = 9 gives two answers rather than one, and why an absolute value equation almost always has two solutions: |a| = |b| means a = ±b, never just a = b.

WHAT TO WORK OUT

|x+|=

|x − 3| = 2

1 and 5

The inside can be either 2 or −2, and both give a valid answer. That is the whole reason an absolute value equation has two solutions and an ordinary linear one has a single answer.

Both cases, worked through. Setting the inside equal to 2 gives x − 3 = 2, so x = 5. Setting it equal to −2 gives x = 1. Both satisfy the original equation, which is easy to check: the inside comes to 2 and -2, and both have absolute value 2. Note that the case setting the inside to +2 gives the larger of the two roots here, because a is positive — with the opposite sign they swap over, so the cases are not in the order the answer is listed.

Step	Working	Result
Inside = +2	x − 3 = 2	x = 5
Inside = −2	x − 3 = −2	x = 1
Check	\|inside\| at each root	2, 2

THE PICTURE

A horizontal line crosses a V in two places, which is the two solutions made visible. Lower the line to the vertex and the two crossings merge into one; lower it below the axis and there are none, because the V never goes there.

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HOW TO USE

1
Choose what you are actually asking. Solving an equation, evaluating at a point, measuring a distance and taking a complex modulus are four different problems that share a notation.
2
For an equation, check the sign of the right-hand side before doing anything. A negative there means no solution, and setting up the two cases anyway produces two numbers that satisfy neither.
3
Read the branch that was used when evaluating. The definition has two, and knowing which one applied is what makes the minus sign in |x| = −x for negative x stop looking wrong.
4
Use the graph to see why an equation has two answers. A horizontal line crosses a V twice, touches it once at the vertex, and misses it entirely below the axis.

REFERENCE FORMULAS

Rule	Formula	What it is for
Definition	\|x\| = x if x ≥ 0, −x if x < 0	Two cases, not one. The minus on the second branch makes the result positive.
Square root	√(x²) = \|x\|	Not x. At x = −3 the left side is 3. This is why x² = 9 gives x = ±3.
Equality	\|a\| = \|b\| ⟺ a = b or a = −b	Never just a = b. This is why an absolute value equation usually has two answers.
Equation	\|u\| = c ⟹ u = c or u = −c	Two solutions when c > 0, one when c = 0, none when c < 0.
Distance	\|a − b\|	The gap between a and b on the number line. Symmetric, since \|−u\| = \|u\|.
Multiplication	\|ab\| = \|a\| · \|b\|	Absolute value passes straight through a product, and through a quotient too.
Addition	\|a + b\| ≤ \|a\| + \|b\|	The triangle inequality. Equality only when a and b share a sign, or one is zero.
Reverse triangle	\| \|a\| − \|b\| \| ≤ \|a − b\|	A lower bound on the distance. Useful for showing two quantities cannot be far apart.
Powers	\|xⁿ\| = \|x\|ⁿ	And for even n, xⁿ is already non-negative, so the bars change nothing.
Sign	x = sgn(x) · \|x\|	Magnitude and direction separated. The absolute value keeps the size and discards the sign.
Complex modulus	\|a + bi\| = √(a² + b²)	The same idea in two dimensions — the distance from the origin.
Not distributive	\|a + b\| ≠ \|a\| + \|b\|	In general. \|3 + (−5)\| is 2, while \|3\| + \|−5\| is 8.
Vertex	y = \|ax + b\| turns at x = −b/a	The graph is a V. The turning point is where the inside changes sign.
Inequalities	\|u\| < c is an AND; \|u\| > c is an OR	Covered on the inequality solver, where the two cases are worked through.

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Last updated: August 4, 2026 · √(x²) = |x|, not x · |a| = |b| means a = ±b, which is why these equations have two answers.