Chapter 3 Section 5

Graphing Linear Inequalities in Two Variables 3.5 Graphing Linear Inequalities in Two Variables Graph linear inequalities in two variables. Graph an inequality with a boundary line through the origin. 2

Graphing Linear Inequalities in Two Variables In Section 3.2, we graphed linear equations such as 2x + 3y = 6. Now we extend this work to linear inequalities in two variables, such as 2x + 3y ≤ 6. Linear Inequality in Two Variables An inequality that can be written as or where A, B, and C are real numbers and A and B are both not 0, is a linear inequality in two variables. Slide 3.5-3

Graph linear inequalities in two variables. Objective 1 Graph linear inequalities in two variables. Slide 3.5-4

Graph linear inequalities in two variables. The graph of the line x + y = 5, called the boundary line, divides the points in the rectangular coordinate system into three sets. 1. Those points that lie on the line itself and satisfy the equation x + y = 5. 2. Those that lie in the region above the line and satisfy the inequality x + y > 5. 3. Those that lie in the region below the line and satisfy the inequality x + y < 5. Graphs of linear inequalities in two variables are regions in the real number plane that may or may not include boundary lines. Slide 3.5-5

Graph linear inequalities in two variables. (cont’d) The inequality 2x + 3y ≤ 6 means that or The graph of 2x + 3y = 6 is a line with intercepts (0, 2). This boundary line divides the plane into two regions, one of which satisfies the inequality. A test point gives a quick way to find the correct region to shade. We choose any point not on the boundary line and substitute it into the given inequality to see whether the resulting statement is true or false. Because (0,0) is easy to substitute into an inequality, it is often a good choice. True Since the last statement is true, we shade the region that includes the test point (0,0). Slide 3.5-6

Graphing a Linear Inequality EXAMPLE 1 Graphing a Linear Inequality Graph 4x − 5y ≤ 20. Solution: The graph should include a solid line since there is an equal sign portion in the equation. True Since the statement is true, the region including the test point should be shaded. Slide 3.5-7

Graph linear inequalities in two variables. Alternately, we can find the required region by solving the inequality for y. Ordered pairs in which y is equal to are on the boundary line, so pairs in which y is less than will be below that line. As we move down vertically, the y-values decrease. This gives the same region that we shaded in the previous method. Slide 3.5-8

Graphing a Linear Inequality EXAMPLE 2 Graphing a Linear Inequality Graph 3x + 5y >15. Solution: The graph should include a dotted line since there is no equal sign in the equation. False Since the statement is false the region above the dotted line should be shaded. Slide 3.5-9

Graphing a Linear Inequality Graph linear inequalities in two variables. (cont’d) Graphing a Linear Inequality Step 1: Graph the boundary. Graph the line that is the boundary of the region. Use the methods of Section 3.2. Draw a solid line if the inequality has ≤ or ≥ because of the equality portion of the symbol. Draw a dashed line if the inequality has < or >. Step 2: Shade the appropriate region. Use any point not on the line as a test point. Substitute for x and y in the inequality. If a true statement results, shade the side containing the test point. If a false statement results, shade the other side. Slide 3.5-10

EXAMPLE 3 Graph y < 4. Solution: Graphing a Linear Inequality with a Vertical Boundary Line Graph y < 4. Solution: The graph should include a dotted line since there is no equal sign in the equation. Since the statement is true the region beow the dotted line should be shaded. Slide 3.5-11

Graph an inequality with a boundary line through the origin. Objective 2 Graph an inequality with a boundary line through the origin. Slide 3.5-12

EXAMPLE 4 Graph x ≥ −3y. Solution: Graphing a Linear Inequality with a Boundary Line through the Origin Graph x ≥ −3y. Solution: The graph should include a solid line since there is an equality in the equation. Test point (1,1) is used. Since the statement is true the region above the line should be shaded. If the graph of an inequality has a boundary line that goes through the origin, (0,0) cannot be used as a test point. Slide 3.5-13