System of Linear Inequalities

System of Linear Inequalities

How to solve the system of linear inequalities problems by graphing the inequalities: examples and their solutions.

Example 1

Solve the given system of linear inequalities. x + y > 2, 2x - y >= 1

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

See x + y > 2.

Move x to the right side.

Then y > -x + 2.

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

Move 2x to the right side.

Multiply (-) on both sides.
This is an inquality.
So, by multiplying (-) on both sides,
the order of the inequality changes.

So y ≤ 2x - 1.

Linear inequalities (One variable) - Example 2

So the system of linear inequalities is

y > -x + 2
y ≤ 2x - 1.

First, graph y > -x + 2
on the coordinate plane.

Graphing linear inequalities on a coordinate plane

Start from the y-intercept: +2.

The slope is -1.
So move 1 unit to the right
and move 1 unit downward.
Let's call this point the 'endpoint'.

[ > ] doesn't include [ = ].
So draw a dashed line
that passes through
the y-intercept and the endpoint.

y [is greater than] the right side.
(y [ > ] -x + 2)

So lightly color the upper region of the dashed line.

Next, 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 move 2 units upward.
Let's call this point the 'endpoint'.

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

y [is lesser than or equal to] the right side.
(y [ ≤ ] 2x - 1)

So lightly color the lower region of the solid line.

The solution of the system is the region
that satisfies both of the inequalities.

So color the intersecting region,
including the boundaries (solid or dashed lines).

Example 2

Solve the given system of linear inequalities. 2x - y > -3, 2x - y >= 1

Change 2x - y > -3
into slope-intercept form.

Move 2x to the right side.

Multiply (-) on both sides.
By multiplying (-) on both sides,
the order of the inequality changes.

So y < 2x + 3.

Linear inequalities (One variable) - Example 2

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

Move 2x to the right side.

Multiply (-) on both sides.
By multiplying (-) on both sides,
the order of the inequality changes.

So y ≤ 2x - 1.

So the system of linear inequalities is

y < 2x + 3
y ≤ 2x - 1.

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

Graphing linear inequalities on a coordinate plane

Start from the y-intercept: +3.

The slope is 2.
So move 1 unit to the right
and move 2 units upward.
Let's call this point the 'endpoint'.

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

y [is lesser than] the right side.
(y [ < ] 2x + 3)

So lightly color the lower region of the dashed line.

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 move 2 units upward.
Let's call this point the 'endpoint'.

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

y [is lesser than or equal to] the right side.
(y [ ≤ ] 2x - 1)

So lightly color the lower region of the solid line.

Color the intersecting region,
including the boundary (solid line).

As you can see,
the second region is in the first region.

So the colored region is the second region.

Example 3

Solve the given system of linear inequalities. 3x + y >= 4, 3x + y < -2

Change 3x + y ≥ 4
into slope-intercept form.

Move 3x to the right side.

Then y ≥ -3x + 4.

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

Move 3x to the right side.

Then y < -3x - 2.

So the system of linear inequalities is

y ≥ -3x + 4
y < -3x - 2.

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

Graphing linear inequalities on a coordinate plane

Start from the y-intercept: +4.

The slope is -3.
So move 1 unit to the right
and move 3 units downward.
Let's call this point the 'endpoint'.

[ ≥ ] does include [ = ].
So draw a solid line
that passes through
the y-intercept and the endpoint.

y [is greater than or equal to] the right side.
(y [ ≥ ] -3x + 4)

So lightly color the upper region of the solid line.

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

Start from the y-intercept: -2.

The slope is -3.
So move 1 unit to the right
and move 3 units downward.
Let's call this point the 'endpoint'.

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

y [is lesser than] the right side.
(y [ < ] -3x - 2)

So lightly color the lower region of the dashed line.

As you can see,
there's no intersecting region.

So this system of inequalities has no solution.