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.