The absolute value function

Absolute value as distance from zero, its piecewise definition, graphs of y = a|x − h| + k, solving absolute value equations, and useful identities.

Absolute value as distance

$|x|$ is the distance of $x$ from $0$ on the number line, so it is never negative: $|5| = 5$ and $|-5| = 5$. More generally, $|a - b|$ is the distance between $a$ and $b$.

Piecewise definition

$$|x| = \begin{cases} x, & x \ge 0 \\ -x, & x < 0 \end{cases}$$

Graphs

$y = |x|$ is a V shape with vertex at the origin, made of $y = x$ (for $x \ge 0$) and $y = -x$ (for $x < 0$). It is an even function. Translations and stretches work as usual: $y = a|x - h| + k$ has its vertex at $(h, k)$.

Absolute value graphs and equations

The intersections of the V with the dashed line are the solutions of a|x − h| + k = c.

Try this

Find a line $y = c$ that meets the V exactly once.
Use the graph to solve $|x - 1| - 2 = 3$, then check algebraically.

Solving absolute value equations

Example 1●

Solve $|2x - 1| = 7$.

$2x - 1 = 7$ or $2x - 1 = -7$

$x = 4$ or $x = -3$

Example 2●●

Sketch $y = |x + 2| - 3$, showing intercepts.

Vertex $(-2, -3)$, V opening up.

$y$-intercept: $|2| - 3 = -1$.

$x + 2 = \pm 3$, so $x = 1$ or $x = -5$.

Set y = 0: |x + 2| = 3.

Useful identities

$|x| = \sqrt{x^2}$, so $\sqrt{x^2} = x$ only when $x \ge 0$.
$|ab| = |a|\,|b|$ and $\left|\dfrac{a}{b}\right| = \dfrac{|a|}{|b|}$.
$|-x| = |x|$ and $|x|^2 = x^2$.
$|a + b| \le |a| + |b|$ (the triangle inequality).

Flashcards

|x| meaning

Distance of $x$ from $0$.

|a − b| meaning

Distance between $a$ and $b$.

|X| = c

$X = c$ or $X = -c$ (no solution if $c < 0$).

√(x²)

$|x|$

Practice

Evaluate $|3 - 10|$.
Solve $|x - 3| = 5$. Enter both solutions.
$|x + 4| = -2$ has:
The vertex of $y = 2|x - 3| + 1$ is:
$\sqrt{x^2}$ equals:
How far apart are $-2$ and $3$ on the number line? (Use $|a - b|$.)
Which is not always true?
Find the $y$-intercept of $y = |x - 4| - 5$.