Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Graphs of Linear Equations 11 11.1 Graphs and Applications of Linear Equations 11.2 More with Graphing and Intercepts 11.3 Slope and Applications 11.4 Equations of Lines 11.5 Graphing Using the Slope and the y-Intercept 11.6 Parallel and Perpendicular Lines 11.7 Graphing Inequalities in Two Variables Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 2

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.7 Graphing Inequalities in Two Variables a Determine whether an ordered pair of numbers is a solution of an inequality in two variables. b Graph linear inequalities. d Use < or > for  to write a true statement in a situation like 6  10. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 3

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.7 Graphing Inequalities in Two Variables a Determine whether an ordered pair of numbers is a solution of an inequality in two variables. A graph of an inequality is a drawing that represents its solutions. An inequality in one variable can be graphed on the number line. An inequality in two variables can be graphed on a coordinate plane. The solutions of inequalities in two variables are ordered pairs. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 4

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.7 Graphing Inequalities in Two Variables a Determine whether an ordered pair of numbers is a solution of an inequality in two variables. 1 Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 5

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.7 Graphing Inequalities in Two Variables b Graph linear inequalities. 3 We first graph the line y = x. Every solution of y = x is an ordered pair like (3, 3) in which the first and second coordinates are the same. We draw the line y = x dashed because its points are not solutions of y > x. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 6

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.7 Graphing Inequalities in Two Variables b Graph linear inequalities. 3 Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 7

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.7 Graphing Inequalities in Two Variables b Graph linear inequalities. 3 Several ordered pairs are plotted in the half-plane above the line y = x. Each is a solution of y > x. We can check a pair such as (–2, 4) as follows: Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 8

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.7 Graphing Inequalities in Two Variables b Graph linear inequalities. 3 It turns out that any point on the same side of y = x as (–2, 4) is also a solution. If we know that one point in a half-plane is a solution, then all points in that half-plane are solutions. We could have chosen other points to check. The graph of y > x is shown on the next slide. (Solutions are indicated by color shading throughout.) We shade the half-plane above y = x. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 9

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.7 Graphing Inequalities in Two Variables b Graph linear inequalities. 3 Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 10

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.7 Graphing Inequalities in Two Variables b Graph linear inequalities. A linear inequality is one that we can get from a linear equation by changing the equals symbol to an inequality symbol. Every linear equation has a graph that is a straight line. The graph of a linear inequality is a halfplane, sometimes including the line along the edge. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 11

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.7 Graphing Inequalities in Two Variables To graph an inequality in two variables: 1. Replace the inequality symbol with an equals sign and graph this related linear equation. 2. If the inequality symbol is < or > draw the line dashed. If the inequality symbol is or , draw the line solid. b Graph linear inequalities. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 12

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.7 Graphing Inequalities in Two Variables 3. The graph consists of a half-plane, either above or below or left or right of the line, and, if the line is solid, the line as well. To determine which half-plane to shade, choose a point not on the line as a test point. Substitute to find whether that point is a solution of the inequality. If it is, shade the half-plane containing that point. If it is not, shade the half-plane on the opposite side of the line. b Graph linear inequalities. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 13

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.7 Graphing Inequalities in Two Variables b Graph linear inequalities. 5 1. First, we graph the line 2x + 3y = 6. The intercepts are (0, 2) and (3, 0). 2. Since the inequality contains the symbol, we draw the line solid to indicate that any pair on the line is a solution. 3. Next, we choose a test point that is not on the line. We substitute to determine whether this point is a solution. The origin is generally an easy point to use: Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 14

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.7 Graphing Inequalities in Two Variables b Graph linear inequalities. 5 We see that (0, 0) is a solution, so we shade the lower half-plane. Had the substitution given us a false inequality, we would have shaded the other half-plane. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 15

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.7 Graphing Inequalities in Two Variables b Graph linear inequalities. 6 There is no y-term in this inequality, but we can rewrite this inequality as x + 0y < 3. We use the same technique that we have used with the other examples. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 16

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.7 Graphing Inequalities in Two Variables b Graph linear inequalities. 6 1. We graph the related equation x = 3 on the plane. 2. Since the inequality symbol is < we use a dashed line. 3. The graph is a half-plane either to the left or to the right of the line x = 3. To determine which, we consider a test point, (–4, 5): Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 17

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.7 Graphing Inequalities in Two Variables b Graph linear inequalities. 6 We see that (–4, 5) is a solution, so all the pairs in the half-plane containing (–4, 5) are solutions. We shade that half-plane. The solutions are all those ordered pairs whose first coordinates are less than 3. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 18

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.7 Graphing Inequalities in Two Variables b Graph linear inequalities. 7 1. We first graph y = –4. 2. We use a solid line to indicate that all points on the line are solutions. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 19

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.7 Graphing Inequalities in Two Variables b Graph linear inequalities. 7 3. We then use (2, 3) as a test point and substitute: Since (2, 3) is a solution, all points in the half-plane containing (2, 3) are solutions. Note that this half-plane consists of all ordered pairs whose second coordinate is greater than or equal to –4. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 20