System of Linear Inequalities

How to graph a system of linear inequalities on the coordinate plane: 3 examples and their solutions.

Example 1

Example

Solution

To graph the linear inequalities,
change the linear inequality to slope-intercept form.

Start from the first inequality x + y > 2.
The slope-intercept form is y > -x + 2.

Change 2x - y ≥ 3 to slope-intercept form.

Move 2x to the right side.
Then -y ≥ -2x + 3.

Divide both sides by -1.
Then y ≤ 2x - 3.
The divisor -1 is minus.
So the order of the inequality sign changes:
from ≥ to ≤.

Linear Inequality (One Variable)

So the given linear inequalities in slope-intercept form are
y > -x + 2 and y ≤ 2x - 3.

Let's graph these inequalities on the coordinate plane.

Graph y > -x + 2 on the coordinate plane.

Start from the y-intercept +2.

The slope is -1.
So move 1 unit to the right and 1 unit downward.
Mark the endpoint.

The inequality sign [>] doesn't include [=].
So draw a dashed line that passes through
the y-intercept +2 and the marked endpoint.

See y > -x + 2.
y is greater than the right side.
So color the upper region of the dashed line.

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

Graph y ≤ 2x - 3 on the coordinate plane.

Start from the y-intercept -3.

The slope is 2.
So move 1 unit to the right and 2 units upward.
Mark the endpoint.

The inequality sign [≤] does include [=].
So draw a solid line that passes through
the y-intercept -3 and the marked endpoint.

See y ≤ 2x - 3.
y is less than (or equal to) the right side.
So color the lower region of the solid line.

This is the graph of y ≤ 2x - 3.

Color the intersecting region.

So the colored region is the answer.

Example 2

Example

Solution

To graph the linear inequalities,
change the linear inequality to slope-intercept form.

Start from the first inequality 2x - y > -3.

Move 2x to the right side.
Then -y > -2x - 3.

Divide both sides by -1.
Then y < 2x - 3.
The divisor -1 is minus.
So the order of the inequality sign changes:
from > to <.

Change 2x - y ≥ 1 to slope-intercept form.

Move 2x to the right side.
Then -y ≥ -2x + 1.

Divide both sides by -1.
Then y ≤ 2x - 1.
The divisor -1 is minus.
So the order of the inequality sign changes:
from ≥ to ≤.

So the given linear inequalities in slope-intercept form are
y < 2x + 3 and y ≤ 2x - 1.

Let's graph these inequalities on the coordinate plane.

Graph y < 2x + 3 on the coordinate plane.

Start from the y-intercept +3.

The slope is 2.
So move 1 unit to the right and 2 units upward.
Mark the endpoint.

The inequality sign [<] doesn't include [=].
So draw a dashed line that passes through
the y-intercept +3 and the marked endpoint.

See y < 2x + 3.
y is less than the right side.
So color the lower region of the dashed line.

This is the graph of y < 2x + 3.

Graph y ≤ 2x - 1 on the coordinate plane.

Start from the y-intercept -1.

The slope is 2.
So move 1 unit to the right and 2 units upward.
Mark the endpoint.

The inequality sign [≤] does include [=].
So draw a solid line that passes through
the y-intercept -1 and the marked endpoint.

See y ≤ 2x - 1.
y is less than (or equal to) the right side.
So color the lower region of the solid line.

This is the graph of y ≤ 2x - 1.

Color the intersecting region.

The region of y < 2x + 3 includes y ≤ 2x - 1.
So color the region of y ≤ 2x - 1.

So the colored region is the answer.

Example 3

Example

Solution

To graph the linear inequalities,
change the linear inequalities to slope-intercept form.

The slope-intercept form of 3x + y ≥ 4
is y ≥ -3x + 4.

Change 3x + y < -2 to slope-intercept form.

Then y < -3x - 2.

So the given linear inequalities in slope-intercept form are
y ≥ -3x + 4 and y < -3x - 2.

Let's graph these inequalities on the coordinate plane.

Graph y ≥ -3x + 4 on the coordinate plane.

Start from the y-intercept +4.

The slope is -3.
So move 1 unit to the right and 3 units downward.
Mark the endpoint.

The inequality sign [≥] does include [=].
So draw a solid line that passes through
the y-intercept +4 and the marked endpoint.

See y ≥ -3x + 4.
y is greater than (or equal to) the right side.
So color the upper region of the solid line.

This is the graph of y ≥ -3x + 4.

Graph y < -3x - 2 on the coordinate plane.

Start from the y-intercept -3.

The slope is -3.
So move 1 unit to the right and 3 units downward.
Mark the endpoint.

The inequality sign [<] doesn't include [=].
So draw a dashed line that passes through
the y-intercept -2 and the marked endpoint.

See y < -3x - 2.
y is less than the right side.
So color the lower region of the dashed line.

This is the graph of y < -3x - 2.

Color the intersecting region.
But there's no intersecting region.

So this system has no solution.