# 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.