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 $y = 2$.

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 $(-2, 2)$.

The standard way to remove an absolute value is by cases. For $x \geq 0$, $|x| = x$, so $x = 2$; for $x < 0$, $|x| = -x$, so $-x = 2$, that is $x = -2$. Each fits the condition of its own range, so both are valid.

Sliding the line up and down changes the number of intersections. For $y = |x|$ and a horizontal line $y = k$: $k > 0$ gives 2 points, $k = 0$ gives the single point at the origin, and $k < 0$ gives none. The large dots on the graph are the intersection points $(2, 2)$ and $(-2, 2)$.