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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