Graphing Linear Inequalities in Two Variables GOAL: graph a linear inequality in two variables

Solution of Linear Inequalities Expressions of the type x + 2y ≤ 8 and 3x – y > 6 are called linear inequalities in two variables. A solution of a linear inequality in two variables is an ordered pair (x, y) which makes the inequality true. Example: (1, 3) is a solution to x + 2y ≤ 8 since (1) + 2(3) = 7 ≤ 8. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Solution of Linear Inequalities

Step 1: Put into slope intercept form Using What We Know Sketch a graph of x + y < 3 Step 1: Put into slope intercept form y <-x + 3 Step 2: Graph the line y = -x + 3

Less than means to the left or below. To check it, pick any point that is not on the line. (0,0) is an easy point to use. x + y < 3 Substitute 0 for x and y. 0 + 0 < 3 0 < 3 Decide if this is true or false. Is 0 less than 3? If it is true, you shade on the same side of the line of the point you picked. If it is false, you shade on the opposite side of the line where the point you picked lies.

Graphing Linear Inequalities The graph of a linear inequality in two variables is the graph of all solutions of the inequality. The boundary line of the inequality divides the coordinate plane into two half-planes: a shaded region which contains the points that are solutions of the inequality, and an unshaded region which contains the points that are not. GRAPHING A LINEAR INEQUALITY The graph of a linear inequality in two variables is a half-plane. To graph a linear inequality, follow these steps. 1 STEP Graph the boundary line of the inequality. Use a dashed line for < or > and a solid line for £ or ³. 2 STEP To decide which side of the boundary line to shade, test a point not on the boundary line to see whether it is a solution of the inequality. Then shade the appropriate half-plane.

Graphing an Inequality To graph the solution set for a linear inequality: 1. Graph the boundary line. 2. Select a test point, not on the boundary line, and determine if it is a solution. 3. Shade a half-plane. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Graphing an Inequality

Graphing a Linear Inequality Sketch a graph of y  3

Graphing an Inequality in Two Variables Graph x < 2 Step 1: Start by graphing the line x = 2 Now what points would give you less than 2? Since it has to be x < 2 we shade everything to the left of the line.

Graph a) y < –2 and b) x £ 1 in a coordinate plane. SOLUTION Graph the boundary line x = 1. Use a solid line because x £ 1. Graph the boundary line y = –2. Use a dashed line because y < – 2. Test the point (0, 0). Because (0, 0) is a solution of the inequality, shade the half-plane to the left of the line. Test the point (0, 0). Because (0, 0) is not a solution of the inequality, shade the half-plane below the line.

Some Helpful Hints If the sign is > or < the line is dashed If the sign is  or  the line will be solid When dealing with just x and y. If the sign > or  the shading either goes up or to the right If the sign is < or  the shading either goes down or to the left

When dealing with slanted lines If it is > or  then you shade above If it is < or  then you shade below the line

2. Shade in the intersection of the half-planes. The set of all solutions of a system of linear inequalities is called its solution set. To graph the solution set for a system of linear inequalities in two variables: 1. Shade the half-plane of solutions for each inequality in the system. 2. Shade in the intersection of the half-planes. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Solution Set

Example: Graph a System of Two Inequalities Example: Graph the solution set for the system of linear inequalities: x y -2x + 3y ≥ 6 Graph the two half-planes. 2 The two half-planes do not intersect; therefore, the solution set is the empty set. 2x – 3y ≥ 12 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Graph a System of Two Inequalities

Example 1 Which ordered pair is a solution of 5x - 2y ≤ 6? (0, -3) (5, 5) (1, -2) (3, 3) ANSWER: A. (0, -3)

Example 2 Graph the inequality x ≤ 4 in a coordinate plane. HINT: Remember HOY VEX. Decide whether to use a solid or dashed line. Use (0, 0) as a test point. Shade where the solutions will be. y x 5 -5

Example 3 Graph 3x - 4y > 12 in a coordinate plane. Sketch the boundary line of the graph. Find the x- and y-intercepts and plot them. Solid or dashed line? Use (0, 0) as a test point. Shade where the solutions are. y x 5 -5

Example 4: Using a new Test Point Graph y < 2/5x in a coordinate plane. Sketch the boundary line of the graph. Find the x- and y-intercept and plot them. Both are the origin! Use the line’s slope to graph another point. Solid or dashed line? Use a test point OTHER than the origin. Shade where the solutions are. y x 5 -5

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