 # Graphing Absolute Value Inequalities How to graph the given absolute value inequalities on the coordinate plane: examples and their solutions.

## Example 1: Graph y > |x| - 2 First draw y = |x| - 2.

Graphing absolute value functions - Example 2

|x| can be either
x (x ≥ 0)
or -x (x < 0).

Solve |x| for these two cases.

Case 1: x ≥ 0

Then |x| = x.

So y = |x| - 2 is
y = x - 2.

Case 2: x < 0

Then |x| = -x.

So y = |x| - 2 is
y = -x - 2.

So y = |x| can be written
as a piecewise function:

y = x - 2 (x ≥ 0)
= -x - 2 (x < 0).

Draw y = |x| - 2.

The inequality sign of [y > |x| - 2]
doesn't include [=].
So use a dashed line.

Graphing piecewise functions

y > |x| - 2

This shows that
y is greater than the right side.

So color the upper region of the dashed line.

## Example 2: Graph |y| ≤ -|x| + 4 First draw |y| = -|x| + 4.

When x ≥ 0 and y ≥ 0,

|y| = -|x| + 4 is
y = -x + 4.

So draw y = -x + 4

The inequality sign of |y| ≤ -|x| + 4
does include [=].
So use a solid line.

Graphing absolute value functions - Example 5

Draw the image of y = -x + 4
under the reflection in the y-axis

Draw the image of y = -x + 4
under the reflection in the origin

Draw the image of y = -x + 4
under the reflection in the x-axis

This is the graph of |y| = -|x| + 4.

|y| ≤ -|x| + 4

This shows that
y is less than (or equal to) the right side.