Intersection of an absolute-value graph and a line

The absolute-value graph $y = |x|$ is a V shape with a corner at the origin. It joins two half-lines: $y = x$ for $x \geq 0$ and $y = -x$ for $x < 0$. We find where this $y = |x|$ meets the line $y = 2$.

Finding the intersections

At an intersection $|x| = 2$. Two numbers have absolute value $2$, namely $2$ and $-2$, so $x = 2, -2$. Both give $y = 2$, so the intersection points are $(2, 2)$ and $(-2, 2)$.

Removing the absolute value by cases

The standard way to remove an absolute value is to split into cases.

That last check matters: each solution must lie in the range it came from, and a value that falls outside its own range has to be thrown away.

Reading it as a distance

The absolute value is also a distance from the origin. Reading $|x| = 2$ as asking for the numbers lying $2$ away from $0$, the number line offers one on the right and one on the left. The two intersections are exactly that left-right symmetry. Saying that $y = |x|$ is even and symmetric about the $y$-axis, and saying that the intersections sit symmetrically at $x = \pm 2$, are the same statement.

Care when squaring both sides

You may also square both sides, turning $|x| = 2$ into $x^2 = 4$ and hence $x = \pm 2$. But squaring discards the sign, so it needs care. The equation $|x| = -1$ obviously has no solution, since the left side is never negative, yet squaring produces $x^2 = 1$ and the false solutions $x = \pm 1$. Whenever you square, substitute back into the original equation to check.

Sliding the line up and down

The corner of the V sits at the origin, and that is where the count changes.

As an inequality

An inequality of the form $|x - a| < d$ describes everything within $d$ of $a$, that is $a - d < x < a + d$. Tolerances on a measurement are written this way precisely because the absolute value is a distance. The large dots on the graph are the intersection points $(2, 2)$ and $(-2, 2)$.