3.7 Graphing Linear Inequalities

Transcription

1 8 CHAPTER Graphs and Functions.7 Graphing Linear Inequalities S Graph Linear Inequalities. Graph the Intersection or Union of Two Linear Inequalities. Graphing Linear Inequalities Recall that the graph of a linear equation in two variables is the graph of all ordered pairs that satisf the equation, and we determined that the graph is a line. Here we graph linear inequalities in two variables; that is, we graph all the ordered pairs that satisf the inequalit. If the equal sign in a linear equation in two variables is replaced with an inequalit smbol, the result is a linear inequalit in two variables. Eamples of Linear Inequalities in Two Variables + Ú To graph the linear inequalit + 6, for eample, we first graph the related boundar equation + =. The resulting boundar line contains all ordered pairs the sum of whose coordinates is. This line separates the plane into two half-planes. All points above the boundar line + = have coordinates that satisf the inequalit + 7, and all points below the line have coordinates that satisf the inequalit + 6. is less than 6 (, ) (0, ) (, ) (, 0) (, ) (, 0) (6, 0) (, ) (, ) (, ) is equal to 6 is greater than The graph, or solution region, for + 6, then, is the half-plane below the boundar line and is shown shaded in the graph on the left. The boundar line is shown dashed since it is not a part of the solution region. These ordered pairs on this line satisf + = and not + 6. The following steps ma be used to graph linear inequalities in two variables. Graphing a Linear Inequalit in Two Variables Step. Step. Step. Graph the boundar line found b replacing the inequalit sign with an equal sign. If the inequalit sign is 6 or 7, graph a dashed line indicating that points on the line are not solutions of the inequalit. If the inequalit sign is or Ú, graph a solid line indicating that points on the line are solutions of the inequalit. Choose a test point not on the boundar line and substitute the coordinates of this test point into the original inequalit. If a true statement is obtained in Step, shade the half-plane that contains the test point. If a false statement is obtained, shade the half-plane that does not contain the test point.

2 Section.7 Graphing Linear Inequalities 8 EXAMPLE Graph Solution First, the boundar line for this inequalit is the graph of - = 6. Graph a dashed boundar line because the inequalit smbol is 6. Net, choose a test point on either side of the boundar line. The point (0, 0) is not on the boundar line, so we use this point. Replacing with 0 and with 0 in the original inequalit leads to the following: Let = 0 and = 0. True Because (0, 0) satisfies the inequalit, so does ever point on the same side of the boundar line as (0, 0). Shade the half-plane that contains (0, 0). The halfplane graph of the inequalit is shown at the right. 6 (0, 0) Ever point in the shaded half-plane satisfies the original inequalit. Notice that the inequalit does not describe a function since its graph does not pass the vertical line test. Graph In general, linear inequalities of the form A + B C, where A and B are not both 0, do not describe functions. EXAMPLE Graph Ú. Solution First, graph the boundar line =. Graph a solid boundar line because the inequalit smbol is Ú. Test a point not on the boundar line to determine which half-plane contains points that satisf the inequalit. We choose (0, ) as our test point. Ú 0 Ú 0 Ú Let = 0 and =. False This point does not satisf the inequalit, so the correct half-plane is on the opposite side of the boundar line from (0, ). The graph of Ú is the boundar line together with the shaded region shown. (0, ) Graph Ú.

3 86 CHAPTER Graphs and Functions CONCEPT CHECK If a point on the boundar line is included in the solution of an inequalit in two variables, should the graph of the boundar line be solid or dashed? Graphing Intersections or Unions of Linear Inequalities The intersection and the union of linear inequalities can also be graphed, as shown in the net two eamples. Recall from Section. that the graph of two inequalities joined b and is the intersection of the graphs of the two inequalities. Also, the graph of two inequalities joined b or is the union of the graphs of the two inequalities. EXAMPLE Graph Ú and Ú -. Solution Graph each inequalit. The word and means the intersection. The intersection of the two graphs is all points common to both regions, as shown b the dark pink shading in the third graph. and Graph the intersection of and -. EXAMPLE Graph + Ú - or -. Solution Graph each inequalit. The word or means the union. The union of the two inequalities is both shaded regions, including the solid boundar lines shown in the third graph. q q Answer to Concept Check: Solid Graph - - or Ú.

4 Section.7 Graphing Linear Inequalities 87 Vocabular, Readiness & Video Check Martin-Ga Interactive Videos Watch the section lecture video and answer the following questions.. From Eample and the lecture before, how do ou find the equation of the boundar line? How do ou determine if the points on the boundar line are solutions to the inequalit?. Based on Eample, describe how ou find the intersection of two linear inequalities in two variables. See Video.7.7 Eercise Set Graph each inequalit. See Eamples and Ú Ú Graph each union or intersection. See Eamples and.. Ú and -. Ú or -. - or Ú 6. - and Ú and and or - Ú 0. - or MIXED Graph each inequalit.. Ú Ú or or 7. + and - 6. Ú and - Ú or Ú or 9. Ú - and 0. Ú - and. + 0 or - 6 Ú. + 0 and - 6 Ú. - 7 and or 7 0 Match each inequalit with its graph Ú + A C B D

5 88 CHAPTER Graphs and Functions Write the inequalit whose graph is given REVIEW AND PREVIEW Evaluate each epression. See Sections. and a b 6. a 7 b Find the domain and the range of each relation. Determine whether the relation is also a function. See Section CONCEPT EXTENSIONS 67. Eplain when a dashed boundar line should be used in the graph of an inequalit. 68. Eplain wh, after the boundar line is sketched, we test a point on either side of this boundar in the original inequalit. Solve. 69. Chris-Craft manufactures boats out of Fiberglas and wood. Fiberglas hulls require hours of work, whereas wood hulls require hours of work. Emploees work at most 0 hours a week. The following inequalities model these restrictions, where represents the number of Fiberglas hulls produced and represents the number of wood hulls produced. Ú 0 c Ú Graph the intersection of these inequalities. 70. Rheem Abo-Zahrah decides that she will stud at most 0 hours ever week and that she must work at least 0 hours ever week. Let represent the hours studing and represent the hours working. Write two inequalities that model this situation and graph their intersection.