# Absolute Value Inequality: Graph

How to graph an absolute value inequality on a coordinate plane: 2 examples and their solutions.

## Example 1

### Example

### Solution

See y > |x| - 2.

There's |x|.

So draw y = x - 2

at x > 0.

(the right side of x = 0).

Absolute Value Function: Graph

The inequality sign does not include [=].

So use a dashed line

when drawing the graph.

Linear Inequality (Two Variables)

There's |x|.

So draw the image of the graph

under the reflection in the y-axis.

To find the region to color,

pick any point

that is not on the graph.

The origin (0, 0) seems to be good.

It'll make the solution simple.

Then put (0, 0)

into the given inequality y > |x| - 2.

Then 0 > |0| - 2.

|0| - 2

= 0 - 2

= -2

So 0 > -2.

Absolute Value

0 > -2 is true.

Then color the region

that includes the picked point (0, 0).

And don't color the region

that is adjacent to the colored region.

(This is the way

to graph the inequalities that are complex.)

This is the graph of y > |x| - 2.

## Example 2

### Example

### Solution

See |y| ≤ -|x| + 4.

There are |x| and |y|.

So draw y = -x + 3

at the quadrant I.

(x > 0 and y > 0).

Absolute Value Function: Graph

The inequality sign does include [=].

So use a solid line

when drawing the graph.

There are |x| and |y|.

So draw the images of the graph

under the reflection

in the x-axis,

in the y-axis,

and in the origin.

To find the region to color,

pick any point

that is not on the graph.

(4, 1) seems to be good.

Put (4, 1)

into the given inequality |y| ≤ -|x| + 4.

Then |1| ≤ -|4| + 4.

|1| = 1

-|4| + 4 = -4 + 4

-4 + 4 = 0

So 1 ≤ 0.

1 ≤ 0 is false.

Then color the region

that does not include the picked point (4, 1).

And don't color the region

that is adjacent to the colored region.

This is the graph of |y| ≤ -|x| + 4.