Absolute value equation

We solve the absolute-value equation $|x - 2| = 3$ by reading it as the intersection of $y = |x - 2|$ and $y = 3$ .

Reading it as a distance

Read the absolute value as a distance and the answer appears without any computation. The expression $|x - 2|$ is the gap between $x$ and $2$ on the number line, so $|x - 2| = 3$ simply asks which numbers lie $3$ away from $2$ . Three to the right gives $5$ ; three to the left gives $-1$ . The solutions are therefore $x = 5$ and $x = -1$ , and the two intersection points are $(5, 3)$ and $(-1, 3)$ . An absolute-value equation is a ruler laid against the number line.

Solving by cases

Range	Equation without the absolute value	Solution	Fits its range
$x \geq 2$	$x - 2 = 3$	$x = 5$	yes
$x < 2$	$-(x - 2) = 3$	$x = -1$	yes

Each candidate fits the range it came from, so both are kept.

Reading the graph

$y = |x - 2|$ is the V of $y = |x|$ shifted $2$ to the right, with its corner at $(2, 0)$ . The height $0$ of that corner is the boundary that decides how often a horizontal line meets the graph.

Line	Number of intersections
$y = k$ with $k > 0$	$2$
$y = 0$	$1$ , the corner itself
$y = k$ with $k < 0$	$0$

An equation such as $|x - 2| = -1$ has no solution precisely because the V never dips below the $x$ -axis.

Moving on to inequalities

The same reading handles inequalities.

Inequality	Meaning as a distance	Solution
$\|x - 2\| < 3$	the gap from $2$ is less than $3$	$-1 < x < 5$
$\|x - 2\| = 3$	the gap is exactly $3$	$x = -1, 5$
$\|x - 2\| > 3$	the gap is more than $3$	$x < -1$ or $x > 5$

On the graph these are the stretches where the V lies below the line $y = 3$ and where it lies above. Writing a measurement tolerance as $|x - a| \leq d$ rests on the same idea of a gap from a center $a$ . The large dots on the graph are the intersections $(5, 3)$ and $(-1, 3)$ .