5 Systems of Equations and Inequalities Copyright Cengage Learning. All rights reserved.
5.5 Systems of Inequalities Copyright Cengage Learning. All rights reserved.
Objectives Graphing an Inequality Systems of Inequalities Systems of Linear Inequalities Application: Feasible Regions 3
Graphing an Inequality 4
Graphing an Inequality We begin by considering the graph of a single inequality. We already know that the graph of y = x 2, for example, is the parabola in Figure 1. If we replace the equal sign by the symbol, we obtain the inequality y x 2 Figure 1 5
Graphing an Inequality Its graph consists of not just the parabola in Figure 1, but also every point whose y-coordinate is larger than x 2. We indicate the solution in Figure 2(a) by shading the points above the parabola. y x 2 Figure 2(a) 6
Graphing an Inequality Similarly, the graph of y x 2 in Figure 2(b) consists of all points on and below the parabola. y x 2 Figure 2(b) 7
Graphing an Inequality However, the graphs of y > x 2 and y < x 2 do not include the points on the parabola itself, as indicated by the dashed curves in Figures 2(c) and 2(d). y > x 2 Figure 2(c) y < x 2 Figure 2(d) 8
Graphing an Inequality The graph of an inequality, in general, consists of a region in the plane whose boundary is the graph of the equation obtained by replacing the inequality sign (,, >, or < ) with an equal sign. 9
Graphing an Inequality To determine which side of the graph gives the solution set of the inequality, we need only check test points. 10
Example 1 Graphs of Inequalities Graph each inequality. (a) x 2 + y 2 < 25 (b) x + 2y 5 Solution: (a) Graph the equation. The graph of the equation x 2 + y 2 = 25 is a circle of radius 5 centered at the origin. 11
Example 1 Solution cont d The points on the circle itself do not satisfy the inequality because it is of the form <, so we graph the circle with a dashed curve, as shown in Figure 3. Graph of x 2 + y 2 < 25 Figure 3 12
Example 1 Solution cont d Graph the inequality. To determine whether the inside or the outside of the circle satisfies the inequality, we use the test points (0, 0) on the inside and (6, 0) on the outside. To do this, we substitute the coordinates of each point into the inequality and check whether the result satisfies the inequality. 13
Example 1 Solution cont d Note that any point inside or outside the circle can serve as a test point. We have chosen these points for simplicity. Our check shows that the points inside the circle satisfy the inequality. A graph of the inequality is shown in Figure 3. Graph of x 2 + y 2 < 25 Figure 3 14
Example 1 Solution cont d (b) Graph the equation. We first graph the equation of x + 2y = 5. The graph is the line shown in Figure 4. Graph of x + 2y 5 Figure 4 15
Example 1 Solution cont d Graph the inequality. Let s use the test points (0, 0) and (5, 5) on either sides of the line. Our check shows that the points above the line satisfy the inequality. 16
Example 1 Solution cont d A graph of the inequality is shown in Figure 4. Graph of x + 2y 5 Figure 4 17
Graphing an Inequality IMPORTANT!! We can write the inequality in Example 1 as From this form of the inequality we see that the solution consists of the points with y-values on or above the line. So the graph of the inequality is the region above the line. 18
Systems of Inequalities 19
Systems of Inequalities We now consider systems of inequalities. The solution set of a system of inequalities in two variables is the set of all points in the coordinate plane that satisfy every inequality in the system. The graph of a system of inequalities is the graph of the solution set. 20
Systems of Inequalities To solve a system of inequalities, we use the following guidelines. 21
Example 2 A System of Two Inequalities Graph the solution of the system of inequalities, and label its vertices (vertices are the intersection points). x 2 + y 2 < 25 x + 2y 5 Solution: These are the two inequalities of Example 1. Here we want to graph only those points that simultaneously satisfy both inequalities. 22
Example 2 Solution cont d Graph each inequality. In Figure 5(a) we graph the solutions of the two inequalities on the same axes (in different colors). x 2 + y 2 < 25 x + 2y 5 Figure 5(a) 23
Example 2 Solution cont d Graph the solution of the system. The solution of the system of inequalities is the intersection of the two graphs. This is the region where the two regions overlap, which is the purple region graphed in Figure 5(b). x 2 + y 2 < 25 x + 2y 5 Figure 5(b) 24
Example 2 Solution cont d Find the Vertices. The points ( 3, 4) and (5, 0) in Figure 5(b) are the vertices of the solution set. They are obtained by solving the system of equations x 2 + y 2 = 25 x + 2y = 5 We solve this system of equations by substitution. 25
Example 2 Solution cont d Solving for x in the second equation gives x = 5 2y, and substituting this into the first equation gives (5 2y) 2 + y 2 = 25 Substitute x = 5 2y (25 20y + 4y 2 ) + y 2 = 25 Expand 20y + 5y 2 = 0 Simplify Thus y = 0 or y = 4. 5y(4 y) = 0 Factor 26
Example 2 Solution cont d When y = 0, we have x = 5 2(0) = 5, and when y = 4, we have x = 5 2(4) = 3. So the points of intersection of these curves are (5, 0) and ( 3, 4). Note that in this case the vertices are not part of the solution set, since they don t satisfy the inequality x 2 + y 2 < 25 (so they are graphed as open circles in the figure). They simply show where the corners of the solution set lie. 27
Systems of Linear Inequalities 28
Systems of Linear Inequalities An inequality is linear if it can be put into one of the following forms: ax + by c ax + by c ax + by > c ax + by < c In the next example we graph the solution set of a system of linear inequalities. 29
Example 3 A System of Four Linear Inequalities Graph the solution set of the system, and label its vertices. x + 3y 12 x + y 8 x 0 y 0 30
Example 3 Solution Graph each inequality. In Figure 6 we first graph the lines given by the equations that correspond to each inequality. To determine the graphs of the first two inequalities, we need to check only one test point. Answer is the entire shaded area. Figure 6(a) Figure 6(b) 31
Example 3 Solution cont d For simplicity let s use the point (0, 0). Since (0, 0) is below the line x + 3y = 12, our check shows that the region on or below the line must satisfy the inequality. 32
Example 3 Solution cont d Likewise, since (0, 0) is below the line x + y = 8, our check shows that the region on or below this line must satisfy the inequality. The inequalities x 0 and y 0 say that x and y are nonnegative. 33
Example 3 Solution cont d These regions are sketched in Figure 6(a). Figure 6(a) 34
Example 3 Solution cont d Graph the solution of the system. The solution of the system of inequalities is the intersection of the graphs. This is the purple region graphed in Figure 6(b). Figure 6(b) 35
Example 3 Solution cont d Find the Vertices. The coordinates of each vertex are obtained by simultaneously solving the equations of the lines that intersect at that vertex. From the system x + 3y = 12 x + y = 8 we get the vertex (6, 2). The origin (0, 0) is also clearly a vertex. The other two vertices are at the x- and y-intercepts of the corresponding lines: (8, 0) and (0, 4). In this case all the vertices are part of the solution set. 36
Systems of Linear Inequalities A region in the plane is called bounded if it can be enclosed in a (sufficiently large) circle. A region that is not bounded is called unbounded. 37
Systems of Linear Inequalities For example, the region graphed in Figure 8 is bounded because it can be enclosed in a circle, as illustrated in Figure 10(a). Figure 8 A bounded region can be enclosed in a circle. Figure 10(a) 38
Systems of Linear Inequalities But the regions graphed in Figure 4 and 10b are unbounded, because we cannot enclose either of them in a circle as illustrated. The shaded areas go on forever, so you can t enclose them in a circle. Graph of x + 2y 5 Figure 4 An unbounded region cannot be enclosed in a circle. Figure 10(b) 39
Example: Graph the solution of the system of inequalities, and label its vertices, and determine whether the solution set is bounded. 40
Example: This is the same problem we just did, but there is one small change on it. Can you see which inequality changed? How does this affect the shading on the graph? 41
Example (like #63 on assignment) Graph the solution of the system of inequalities, and label its vertices, and determine whether the solution set is bounded. (This example is just like #63.) 42
Assignment: Section 5.5: problems 1-21 odd, 27-63 odd 43