Goal

• Graphing linear inequalities in two variables

Remember the Symbols

Less than

Greater than

Less than OR EQUAL TO

Greater than OR EQUAL TO

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 Which ordered pair is NOT a solution of x - 3y ≤ 6≤ 6??

A. (0,0) B. (6,-1) C. (10, 3) D. (-1,2)A. (0,0) B. (6,-1) C. (10, 3) D. (-1,2)

x - 3y ≤ 6x - 3y ≤ 6

Substitute each point into the inequality. If the statement is true then it is a solution.

(0) – 3(0) ≤ 6(0) – 3(0) ≤ 6 True, therefore(0,0) is a solution.

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.

• There is a BOUNDARY LINE that divides the coordinate plane into two HALF-PLANES.

Only one half-plane contains the points that represent the solutions to the inequality.

Graphing Linear Inequalities

• Graphing Boundary Lines:

• Use a dashed line for < or >.

• Use a solid line for ≤ or ≥.

6

To graph the solution set for a linear inequality:

2. Select a test point, not on the boundary line, usually the origin, (0,0) and determine if it is a solution.

3. Shade a half-plane.

1. Graph the boundary line.

Graph an Inequality

Graph the equation

STEPSTEP 11

Graph the inequality Graph the inequality y > 4x - 3y > 4x - 3..

STEPSTEP 22

Test (0,0) in the original inequality.

STEPSTEP 33

Shade the half-plane that contains the point (0,0), because (0,0) is a solution to the inequality.

34 xy 34 xy

3)0(40

True

Graph an Inequality

Graph the equation

STEPSTEP 11

Graph the inequality Graph the inequality x + 2y ≤ 0..

STEPSTEP 22

Test (1,0) in the original inequality.

STEPSTEP 33Shade the half-plane that does not contain the point (1,0), because (1,0) is not a solution to the inequality.

02 yx 02 yx

0)0(21

False

Graph an Inequality

Graph the equation

STEPSTEP 11

Graph the inequality Graph the inequality x + 3y ≥ -1..

STEPSTEP 22

Test (1,0) in the original inequality.

STEPSTEP 33Shade the half-plane that contains the point (1,0), because (1,0) is a solution to the inequality. 13 yx 13 yx

1)0(31

True

Graph an Inequality

Graph the equation

STEPSTEP 11

Graph the inequality Graph the inequality y ≥ -3..

STEPSTEP 22

Test (2,0) in the original inequality. Use only the y-coordinate, because the inequality does not have a x-variable.

STEPSTEP 33Shade the half-plane that contains the point (2,0), because (2,0) is a solution to the inequality. 3y

3y

3)0(

True

Graph an Inequality

Graph the equation

STEPSTEP 11

Graph the inequality Graph the inequality xx ≤ -1..

STEPSTEP 22

Test (3,0) in the original inequality. Use only the y-coordinate, because the inequality does not have a x-variable.

STEPSTEP 33Shade the half-plane that does not contain the point (3,0), because (3,0) is not a solution to the inequality.

1x

1x

1)0(

False