1. Objective
Graph Linear Equations and Linear Inequalities in Two Variables

2. Section 7.8 “Graph Linear Inequalities”
the result of replacing the = sign
in a linear equation with an inequality sign.

3. Linear Inequalities
An example of a linear inequality in two variables is x - 3y ≤ 6. The solution of an inequality in two variables, x and y, is an ordered pair (x, y) that produces a true statement when substituted into the inequality.
Which ordered pair is NOT a solution of x - 3y ≤ 6?
A. (0,0) B. (6,-1) C. (10, 3) D. (-1,2)
Substitute each point into the inequality. If the statement is true then it is a solution.
True, therefore
(0,0) is a solution.
x - 3y ≤ 6
(0) – 3(0) ≤ 6

4. Graph an Inequality in Two Variables
The graph of an inequality in two variables is the set of points that represent all solutions of the inequality.
The BOUNDARY LINE of a linear inequality divides the coordinate plane into two HALF-PLANES. Only one half-plane contains the points that represent the solutions to the inequality.

5. Graphing Linear Inequalities
Graphing Boundary Lines:
Use a dashed line for < or >.
Use a solid line for ≤ or ≥.

6. Graph the inequality y > 4x - 3.
Graph an Inequality
Graph the inequality y > 4x - 3.
STEP 2
STEP 3
STEP 1
Graph the equation
Test (0,0) in the original inequality.
Shade the half-plane that contains the point (0,0), because (0,0) is a solution to the inequality.

7. Graph the inequality x + 2y ≤ 0.
Graph an Inequality
Graph the inequality x + 2y ≤ 0.
STEP 2
STEP 3
STEP 1
Shade the half-plane that does not contain the point (1,0), because (1,0) is not a solution to the inequality.
Graph the equation
Test (1,0) in the original inequality.

8. Graph the inequality x + 3y ≥ -1.
Graph an Inequality
Graph the inequality x + 3y ≥ -1.
STEP 2
STEP 3
STEP 1
Shade the half-plane that contains the point (1,0), because (1,0) is a solution to the inequality.
Graph the equation
Test (1,0) in the original inequality.

9. Graph the inequality y ≥ -3.
Graph an Inequality
Graph the inequality y ≥ -3.
STEP 2
STEP 3
STEP 1
Shade the half-plane that contains the point (2,0), because (2,0) is a solution to the inequality.
Graph the equation
Test (2,0) in the original inequality. Use only the y-coordinate, because the inequality does not have a x-variable.

10. Graph the inequality x ≤ -1.
Graph an Inequality
Graph the inequality x ≤ -1.
STEP 2
STEP 3
STEP 1
Shade the half-plane that does not contain the point (3,0), because (3,0) is not a solution to the inequality.
Graph the equation
Test (3,0) in the original inequality. Use only the y-coordinate, because the inequality does not have a x-variable.

11. Challenge “Can You Write and Graph the Mystery Inequality???”
The points (2,5) and (-3, -5) lie on the boundary line. The points (6,5) and (-2, -3) are solutions of the inequality.
y ≤ 2x + 1

12. Graph Absolute Value Functions Extension Activity 6.5
g(x) = |x -3|
f(x) = |x|
g(x) = |x - 3| x
f(x)=|x| -5 5 -2 2 1 3 x
g(x)=|x-3| -3 6 3 4 1
f(x) = |x|