Graphing Linear Inequalities

Multi-Step Equations with Variables on Both Sides - How it Works - Video

Example 1 Part 1

Example 1 Part 1:

When graphing linear inequalities, our first step is graph the line. There are more multiple ways of doing that including leaving it standard form and finding the two points, putting in slope intercept form, or something else. Here we left our inequality in standard form and found our x-intercept and y-intercept.

5y - 2x < 5

5 * (0) - 2x < 5

-2x < 5

x > -2.5

(-2.5, 0)

Rewrite the inequality

Substitute 0 for y to solve for x

Multiply

Divide and flip sign

Create point

5y - 2x < 5

5y - 2 * (0) < 5

5y < 5

y < 1

(0, 1)

Rewrite the inequality

Substitute 0 for y to solve for x

Multiply

Divide and flip sign

Create point

Now, we plot the two points and draw our dotted line. Since our two points are on the line, when we plot them, they are open circles because we have less than.

Example 1 Part 2

Example 1 Part 2:

Now we have to shade. So we can pick two numbers, one above the dotted line (3, 3) and below the dotted line (3,1). The one that is true is the one that we keep or shade.

Example 1 Part 3

Example 1 Part 3:

Now we plug in our two points that we picked into the inequality and see which is true.

5y - 2x < 5

5 * (1) - 2 * (3) < 5

5 - 6 < 5

1 < 5

True

Rewrite the inequality

Substitute the point (3, 1)

Multiply

Subtract


5y - 2x < 5

5 * (3) - 2 * (3) < 5

15 - 6 < 5

9 < 5

Not true

Rewrite the inequality

Substitute 0 for y to solve for x

Multiply

Subtract


Since the point (3, 1) makes the inequality true, we shade the region that contains that point.

Example 2 Part 1

Example 2 Part 1:

When graphing linear inequalities, our first step is graph the line. There are more multiple ways of doing that including leaving it standard form and finding the two points, putting in slope intercept form, or something else. Here we left our inequality in standard form and found our x-intercept and y-intercept.

2y + 4x 8

2 * (0) + 4x 8

4x 8

x 2

(2, 0)

Rewrite the inequality

Substitute 0 for y to solve for x

Multiply

Divide

Create point

2y + 4x 8

2y + 4 * (0) 8

2y 8

x 4

(0, 4)

Rewrite the inequality

Substitute 0 for y to solve for x

Multiply

Divide

Create point

Now, we plot the two points and draw our solid line. Since our two points are on the line, when we plot them, they are closed circles because we have greater than or equal to.

Example 2 Part 2

Example 2 Part 2:

Now we have to shade. So we can pick two numbers, one above the dotted line (1, 0) and below the dotted line (3, 0). The one that is true is the one that we keep or shade.

Example 2 Part 3

Example 2 Part 3:

Now we plug in our two points that we picked into the inequality and see which is true.

2y + 4x 8

2 * (0) + 4 * (1) 8

0 + 4 5

4 5

Not true

Rewrite the inequality

Substitute the point (1, 0)

Multiply

Add


2y + 4x 8

2 * (0) + 4 * (3) 8

0 + 12 5

12 5

True

Rewrite the inequality

Substitute the point (3, 0)

Multiply

Add


Since the point (3, 0) makes the inequality true, we shade the region that contains that point.

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