When graphing an absolute value inequality, we need to create two inequalities. One of the inequalities is the same as the original but without the absolute value. The second inequality has both the symbol and the sign of the left-hand side changed.

∣ y ∣ \geq x \Rightarrow {y \geq x y \leq - x (I) (II)

Note that a pair of numbers

(x, y)

satisfies the original inequality if it satisfies either of the two inequalities. This means that the solution set of our inequality will be the union of the solution sets of the two inequalities.

∣ y ∣ \geq x \Leftrightarrow y \geq x or y \leq - x

We now have a system of inequalities to graph. Both of the inequalities are non-strict, so both lines are going to be solid. Let's start by graphing the boundary line for the first inequality.

We will use

(1, 0)

as the test point to determine which region to shade.

y \geq x \Rightarrow 0 ≱ 1 \times

Since substituting the point made the inequality false, we will shade the region that does not contain it.

We will use the same process to graph the second inequality. Let's use the same test point,

(1, 0),

to determine which region to shade.

y \leq - x \Rightarrow 0 ≰ - 1 \times

Since substituting the point made the inequality false, we will once again shade the region that does not contain it.

The solution set of the given inequality is the union of all the shaded regions.

Exercise