# 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.

This graph is the answer.

## 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

on the quadrant I.

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

on the quadrant II.

Draw the image of *y* = -*x* + 4

under the reflection in the origin

on the quadrant III.

Draw the image of *y* = -*x* + 4

under the reflection in the *x*-axis

on the quadrant IV.

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

|*y*| ≤ -|*x*| + 4

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

So, for the quadrant I,

color the lower region of the solid line.

The lower region of the solid line for the quadrant I

is the inner region of the graph.

So color the inner region of the graph.

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