(6.5) Graphing a Linear Inequality

Graphing a linear inequality is

very similar to graphing a linear equation.

Graphing a Linear Inequality

1) Solve the inequality for y

(or for x if there is no y). 2) Change the inequality to an equation

and graph. 3) If the inequality is < or >, the line

is dotted. If the inequality is ≤ or

≥, the line is solid.

y = 2x + 1

Rise

Run

2

1

slope

y-intercept

1

2

run

rise

b = 1

m = 2

y ≤ 2x + 1

y = 2x + 1 Now for the shadingy ≤ 2x + 1

Pick a point on either side of

the graph

Let’s try (2, 1)

Does the point satisfy the inequality?

1 ≤ 2(2) + 1

1 ≤ 4 + 11 ≤ 5

Therefore, shade the half-plane with the point.

y = 2x + 1 What if we picked a point on the other side of the line?y ≤ 2x + 1

Let’s try (-2, 3)

Does the point satisfy the inequality?

3 ≤ 2(-2) + 1

3 ≤ -4 + 13 ≤ -3

Therefore, shade the otherhalf-plane opposite the point.

y = -3x + 2

Rise

Run

-3

1

slope

y-intercept

1

3

run

rise

b = 2

m = -3

y ≥ -3x + 2

y = -3x + 2y ≥ -3x + 2

Now for the shading

Pick a point on either side of

the graph

Let’s try (0, 0)

Does the point satisfy the inequality?

0 ≥ -3(0) + 2

0 ≥ 0 + 2

0 ≥ 2

Therefore, shade the other half-plane opposite the point.

Rise

Run

2

3

slope

y-intercept

3

2

run

rise

b = -1

13

2 xy

3

2m

13

2 xy Now for the

shading

Pick a point on either side of the graph

Let’s try (0, 0)

Does the point satisfy the inequality?

Therefore, shade the half-plane with the point.

103

20

100 10

Graphing LinearInequalities

2x + 3y ≥ 5x-intercept

2x + 3(0) = 52x + 0 = 5

2x = 5

25x

y-intercept

2(0) + 3y = 50 + 3y = 5

3y = 5

35y

35,0

0,25

2x + 3y ≥ 5

Now for the shading

Pick a point on either side of the graph

Let’s try (4, 2)

Does the point satisfy the inequality?

Therefore, shade the half-plane with the point.

5)2(3)4(2 568 514

2x + 3y ≥ 5

Let’s try (-3, -4)

Does the point satisfy the inequality?

Therefore, shade the other half-plane opposite the point.

5)4(3)3(2 5126 518

What if we picked a point on the other side of the line?

4x - y > 6

x-intercept

4x - (0) = 64x - 0 = 5

4x = 6

23x

y-intercept

4(0) - y = 60 - y = 6

-y = 66y

6,0

0,23

4x - y > 6

Now for the shading

Pick a point on either side of the graph

Let’s try (0, 0)

Does the point satisfy the inequality?

Therefore, shade the other half-plane opposite the point.

6)0()0(4 600 60

4x + 2y < 3

x-intercept

4x + 2(0) = 34x + 0 = 3

4x = 3

43x

y-intercept

4(0) + 2y = 30 + 2y = 3

2y = 3

23y

23,0

0,25

4x + 2y < 3

Now for the shading

Pick a point on either side of the graph

Let’s try (0, 0)

Does the point satisfy the inequality?

Therefore, shade the half-plane with the point.

3)0(2)0(4 300 30

3x - 2y < 7

x-intercept

3x - 2(0) = 73x - 0 = 7

3x = 7

37x

y-intercept

3(0) - 2y = 70 - 2y = 7

-2y = 7

27y

27,0

0,37

3x - 2y < 7

Now for the shading

Pick a point on either side of the graph

Let’s try (0, 0)

Does the point satisfy the inequality?

Therefore, shade the half-plane with the point.

7)0(2)0(3 700 70

Graphing a Linear Inequality Graph the inequality 3 - x > 0 First, solve the inequality for x.

3 - x > 0

-x > -3

x < 3

Graph: x<3

Graph the line x = 3.

Because x < 3 and not x ≤ 3, the line will be dotted.

Now shade the side of the line where x < 3 (to the left of the line).

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4

2

3

Graphing a Linear Inequality 4) To check that the shading is correct, pick a point in the area and plug it into the inequality. 5) If the inequality statement is true, the shading is correct. If the inequality statement is false, the shading is incorrect.

Graphing a Linear Inequality Pick a point, (1,2), in the shaded area. Substitute into the original inequality 3 – x > 0 3 – 1 > 0 2 > 0 True! The inequality has been graphed correctly.

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4

2

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