1. Linear Inequalities Objective: to graph linear inequalities and their systems

2. When you replace the equals sign in a linear equation by one of the inequality symbols, you now have a linear inequality.
Examples: 1 2
y >
x + 1
2x – 3y < 12

3. Any ordered pair of numbers that satisfies the inequality is a solution of the inequality.
Just like an inequality in one variable, linear inequalities can have infinitely many solutions.

4. y < x + 1
Run 1
y-intercept
Rise 1
b = 1
slope
m = 1

5. y < x + 1 y = 2x + 1 Shade TRUE 0 < 0 + 1 0 < 1
Now for the shading . . .
Pick a point on either side of the graph.
Let’s try (0, 0):
Shade
Does the point satisfy the inequality?
0 < 0 + 1
TRUE
Therefore, shade the half-plane with that point.
0 < 1

6. y < x + 1 y = 2x + 1 Don't Shade Shade FALSE 2 < - 3 + 1
What if we picked a point on the other side of the line?
Now try (-3, 2):
Does the point satisfy the inequality?
Shade
2 <
2 < - 2
FALSE
Therefore, shade the half-plane opposite that point.

7. 4x - 3y > 12 find the x value when y = 0 find the y value
x-intercept
4x - 3(0) = 12
4x - 0 = 12
4x = 12
x = 3
find the y value
when x = 0
y-intercept
4(0) - 3y = 12
0 - 3y = 12
-3y = 12
y = -4

8. 4x - 3y > 12 Don't Shade Shade FALSE 4(0) - 3(0) > 12
Now for the shading . . .
Pick a point on either side of the graph.
Don't Shade
Let’s try (0, 0):
Does the point satisfy the inequality?
Shade
FALSE
4(0) - 3(0) > 12
0 - 0 > 12
Therefore, shade the other half-plane opposite the point.
0 > 12

9. We can also have systems of linear inequalities.
Just like systems of equations, solutions to systems of linear inequalities must satisfy both inequalities in the system.

10. Which region satisfies the system?
2x + y > 1
y = 2x + 1
x – y < 3
Which region satisfies the system?