10.4 Nonlinear Inequalities and Systems of Inequalities. OBJECTIVES 1 Graph a Nonlinear Inequality. 2 Graph a System of Nonlinear Inequalities.

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1 Section 0. Nonlinear Inequalities and Sstems of Inequalities 6 CONCEPT EXTENSIONS For the eercises below, see the Concept Check in this section.. Without graphing, how can ou tell that the graph of + = and + = do not have an points of intersection?. Without solving, how can ou tell that the graphs of = + and = + 7 do not have an points of intersection?. How man real solutions are possible for a sstem of equations whose graphs are a circle and a parabola? Draw diagrams to illustrate each possibilit.. How man real solutions are possible for a sstem of equations whose graphs are an ellipse and a line? Draw diagrams to illustrate each possibilit. Solve.. The sum of the squares of two numbers is 0. The difference of the squares of the two numbers is. Find the two numbers. 6. The sum of the squares of two numbers is 0. Their product is 8. Find the two numbers. 7. During the development stage of a new rectangular kepad for a securit sstem, it was decided that the area of the rectangle should be 8 square centimeters and the perimeter should be 68 centimeters. Find the dimensions of the kepad. 8. A rectangular holding pen for cattle is to be designed so that its perimeter is 9 feet and its area is feet. Find the dimensions of the holding pen. Recall that in business, a demand function epresses the quantit of a commodit demanded as a function of the commodit s unit price. A suppl function epresses the quantit of a commodit supplied as a function of the commodit s unit price. When the quantit produced and supplied is equal to the quantit demanded, then we have what is called market equilibrium. Demand function Market equilibrium Suppl function 9. The demand function for a certain compact disc is given b the function p = and the corresponding suppl function is given b p = where p is in dollars and is in thousands of units. Find the equilibrium quantit and the corresponding price b solving the sstem consisting of the two given equations. 0. The demand function for a certain stle of picture frame is given b the function p = and the corresponding suppl function is given b p = 9 + where p is in dollars and is in thousands of units. Find the equilibrium quantit and the corresponding price b solving the sstem consisting of the two given equations. Use a graphing calculator to verif the results of each eercise.. Eercise.. Eercise.. Eercise.. Eercise. 0. Nonlinear Inequalities and Sstems of Inequalities S Graph a Nonlinear Inequalit. Graph a Sstem of Nonlinear Inequalities. Graphing Nonlinear Inequalities We can graph a nonlinear inequalit in two variables such as 9 + in a wa 6 similar to the wa we graphed a linear inequalit in two variables in Section.7. First, graph the related equation 9 + =. The graph of the equation is our 6 boundar. Then, using test points, we determine and shade the region whose points satisf the inequalit. EXAMPLE Graph Solution of First, graph the equation (Continued on net page) includes the graph of =. Sketch a solid curve since the graph =. The graph is an ellipse, and it

2 66 CHAPTER 0 Conic Sections divides the plane into two regions, the inside and the outside of the ellipse. To determine which region contains the solutions, select a test point in either region and determine whether the coordinates of the point satisf the inequalit. We choose (0, 0) as the test point Let = 0 and = True Since this statement is true, the solution set is the region containing (0, 0). The graph of the solution set includes the points on and inside the ellipse, as shaded in the figure. 9 6 Graph Ú. EXAMPLE Graph Solution The related equation is = + 6. Subtract from both sides and divide both sides b 6, and we have - =, which is a hperbola. Graph the 6 hperbola as a dashed curve since the graph of does not include the graph of = + 6. The hperbola divides the plane into three regions. Select a test point in each region not on a boundar line to determine whether that region contains solutions of the inequalit. Test Region A with (0, ) Test Region B with (0, 0) Test Region C with (0, ) True False True The graph of the solution set includes the shaded regions A and C onl, not the boundar. Region A Region B Region C 6 Graph Graphing Sstems of Nonlinear Inequalities In Sections.7 and. we graphed sstems of linear inequalities. Recall that the graph of a sstem of inequalities is the intersection of the graphs of the inequalities.

3 Section 0. Nonlinear Inequalities and Sstems of Inequalities 67 EXAMPLE Graph the sstem e - Solution We graph each inequalit on the same set of aes. The intersection is shown in the third graph below. It is the darkest shaded (appears purple) region along with its boundar lines. The coordinates of the points of intersection can be found b solving the related sstem. e = - = (, ) solution region (q, ~) Graph the sstem e Ú - +. EXAMPLE Graph the sstem d Solution We graph each inequalit. The graph of + 6 contains points inside the circle that has center (0, 0) and radius. The graph of 9-6 is the region between the two branches of the hperbola with -intercepts - and and center (0, 0). The graph of 6 + is the region below the line with slope and -intercept (0, ). The graph of the solution set of the sstem is the intersection of all the graphs, the darkest shaded region shown. The boundar of this region is not part of the solution. 9 solution region

4 68 CHAPTER 0 Conic Sections Graph the sstem d Vocabular, Readiness & Video Check Martin-Ga Interactive Videos Watch the section lecture video and answer the following questions.. From Eample, eplain the similarities between graphing linear inequalities and graphing nonlinear inequalities.. From Eample, describe one possible illustration of graphs of two circle inequalities in which the sstem has no solution that is, the graph of the inequalities in the sstem do not overlap. See Video Eercise Set Graph each inequalit. See Eamples and Ú Ú Ú e - Ú Ú 0 + Ú. c d Ú - REVIEW AND PREVIEW 8.. e - Ú Ú c Ú 6. d 6 + Ú Determine whether each graph is the graph of a function. See Section.. Graph each sstem. See Eamples and. + Ú. e e e Ú. e 7 Ú +. e + Ú 9 + Ú 6 6. e e e Ú + Ú + - Ú 9 0. c

5 Chapter 0 Highlights 69 Find each function value if f = -. See Section... f -. f -. fa. fb CONCEPT EXTENSIONS. Discuss how graphing a linear inequalit such as is similar to graphing a nonlinear inequalit such as Discuss how graphing a linear inequalit such as is different from graphing a nonlinear inequalit such as Ú + 7. Graph the sstem d. Ú 0 Ú 0 Ú 0 Ú 0 8. Graph the sstem: d Ú + - Chapter 0 Vocabular Check Fill in each blank with one of the words or phrases listed below. circle ellipse hperbola conic sections verte diameter center radius nonlinear sstem of equations. A(n) is the set of all points in a plane that are the same distance from a fied point, called the.. A(n) is a sstem of equations at least one of which is not linear.. A(n) is the set of points in a plane such that the sum of the distances of those points from two fied points is a constant.. In a circle, the distance from the center to a point of the circle is called its.. A(n) is the set of points in a plane such that the absolute value of the difference of the distance from two fied points is constant. 6. The circle, parabola, ellipse, and hperbola are called the. 7. For a parabola that opens upward, the lowest point is the. 8. Twice a circle s radius is its. Chapter 0 Highlights DEFINITIONS AND CONCEPTS EXAMPLES Section 0. The Parabola and the Circle Parabolas Graph = a - h + k = = - a 0 (h, k) - + = - + Add () to both sides. = - + (h, k) a 0 Since a =, this parabola opens to the right with verte (, ). Its ais of smmetr is =. The -intercept is (, 0). (continued)

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