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 intercepts, 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 intercepts, 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.