Graphing Linear Inequalities

Transcription

1 7.4 Graphing Linear Inequalities 7.4 OBJECTIVE 1. Graph a linear inequalit in two variables In Section 2.7 ou learned to graph inequalities in one variable on a number line. We now want to etend our work with graphing to include linear inequalities in two variables. We begin with a definition. Definitions: Linear Inequalit in Two Variables An inequalit that can be written in the form NOTE The inequalit smbols,, and can also be used. A B C in which A and B are not both 0, is called a linear inequalit in two variables. Some eamples of linear inequalities in two variables are The graph of a linear inequalit is alwas a region (actuall a half plane) of the plane whose boundar is a straight line. Let s look at an eample of graphing such an inequalit. Eample 1 Graphing a Linear Inequalit NOTE The dotted line indicates that the points on the line 2 4 are not part of the solution to the inequalit 2 4. Graph 2 4. First, replace the inequalit smbol ( ) with an equals sign. We then have 2 4. This equation forms the boundar line of the graph of the original inequalit. You can graph the line b an of the methods discussed earlier. The boundar line for our inequalit is shown at left. We see that the boundar line separates the plane into two regions, each of which is called a half plane. We now need to choose the correct half plane. Choose an convenient test point not on the boundar line. The origin (0, 0) is a good choice because it makes for eas calculation. Substitute 0 and 0 into the inequalit. (0, 0) 2 4 NOTE You can alwas use the origin for a test point unless the boundar line passes through the origin A true statement Because the inequalit is true for the test point, we shade the half plane containing that test point (here the origin). The origin and all other points below the boundar line then represent solutions for our original inequalit. 585

2 586 CHAPTER 7 GRAPHING AND INEQUALITIES CHECK YOURSELF 1 Graph the inequalit 3 3. The process is similar when the boundar line is included in the solution. Eample 2 Graphing a Linear Inequalit NOTE Again, we replace the inequalit smbol ( ) with an equals sign to write the equation for our boundar line. Graph First, graph the boundar line, Note: When equalit is included ( or ), use a solid line for the graph of the boundar line. This means the line is included in the graph of the linear inequalit. The graph of our boundar line (a solid line here) is shown on the figure NOTE Although an of our graphing methods can be used here, the intercept method is probabl the most efficient. (0, 0) Again, we use (0, 0) as a convenient test point. Substituting 0 for and for in the original inequalit, we have A false statement Because the inequalit is false for the test point, we shade the half plane that does not contain that test point, here (0, 0). NOTE All points on and below the boundar line represent solutions for our original inequalit. (0, 0) CHECK YOURSELF 2 Graph the inequalit

3 GRAPHING LINEAR INEQUALITIES SECTION Eample 3 Graphing a Linear Inequalit Graph 5. The boundar line is 5. Its graph is a solid line because equalit is included. Using (0, 0) as a test point, we substitute 0 for with the result 0 5 A true statement Because the inequalit is true for the test point, we shade the half plane containing the origin. NOTE If the correct half plane is obvious, ou ma not need to use a test point. Did ou know without testing which half plane to shade in this eample? 5 CHECK YOURSELF 3 Graph the inequalit 2. As we mentioned earlier, we ma have to use a point other than the origin as our test point. Eample 4 illustrates this approach. Eample 4 Graphing a Linear Inequalit Graph The boundar line is Its graph is shown on the figure. NOTE We use a dotted line for our boundar line because equalit is not included. (0, 0) (1, 1)

4 588 CHAPTER 7 GRAPHING AND INEQUALITIES We cannot use (0, 0) as our test point in this case. Do ou see wh? Choose an other point not on the line. For instance, we have picked (1, 1) as a test point. Substituting 1 for and 1 for gives (0, 0) (1, 1) A false statement Because the inequalit is false at our test point, we shade the half plane not containing (1, 1). This is shown in the graph in the margin. CHECK YOURSELF 4 Graph the inequalit 2 0. The following steps summarize our work in graphing linear inequalities in two variables. Step b Step: To Graph a Linear Inequalit Step 1 Step 2 Step 3 Step 4 Replace the inequalit smbol with an equals sign to form the equation of the boundar line of the graph. Graph the boundar line. Use a dotted line if equalit is not included ( or ). Use a solid line if equalit is included ( or ). Choose an convenient test point not on the line. If the inequalit is true at the checkpoint, shade the half plane including the test point. If the inequalit is false at the checkpoint, shade the half plane not including the test point. CHECK YOURSELF ANSWERS

5 Name 7.4 Eercises Section Date In eercises 1 to 8, we have graphed the boundar line for the linear inequalit. Determine the correct half plane in each case, and complete the graph ANSWERS

6 ANSWERS Graph each of the following inequalities

7 ANSWERS

8 ANSWERS

9 ANSWERS

10 ANSWERS Graph each of the following inequalities ( ) ( ) ( ) 3( ) (2 ) 4(2 ) Hours worked. Suppose ou have two part-time jobs. One is at a video store that pas $9 per hour and the other is at a convenience store that pas $8 per hour. Between the two jobs, ou want to earn at least $240 per week. Write an inequalit that shows the various number of hours ou can work at each job. 594

11 ANSWERS 38. Mone problem. You have at least $30 in change in our drawer, consisting of dimes and quarters. Write an inequalit that shows the different number of coins in our drawer Linda Williams has just begun a nurser business and seeks our advice. She has limited funds to spend and wants to stock two kinds of fruit-bearing plants. She lives in the northeastern part of Teas and thinks that blueberr bushes and peach trees would sell well there. Linda can bu blueberr bushes from a supplier for $2.50 each and oung peach trees for $5.50 each. She wants to know what combination she should bu and keep her outla to $500 or less. Write an equation and draw a graph to depict what combinations of blueberr bushes and peach trees she can bu for the amount of mone she has. Eplain the graph and her options. 40. After reading an article on the front page of The New York Times titled You Have to be Good at Algebra to Figure Out the Best Deal for Long Distance, Rafaella De La Cruz decided to appl her skills in algebra to tr to decide between two competing long-distance companies. It was difficult at first to get the companies to eplain their charge policies. The both kept repeating that the were 25% cheaper than their competition. Finall, Rafaella found someone who eplained that the charge depended on when she called, where she called, how long she talked, and how often she called. Too man variables! she eclaimed. So she decided to ask one compan what the charged as a base amount, just for using the service. Compan A said that the charged $5 for the privilege of using their long-distance service whether or not she made an phone calls, and that because of this fee the were able to allow her to call anwhere in the United States after 6 P.M. for onl $0.15 a minute. Complete this table of charges based on this compan s plan: Total Minutes Long Distance in 1 Month (After 6 P.M.) 0 minutes 10 minutes 30 minutes 60 minutes 120 minutes Total Charge Use this table to make a whole-page graph of the monthl charges from Compan A based on the number of minutes of long distance. 595

12 ANSWERS a. b. c. d. e. Rafaella wanted to compare this offer to Compan B, which she was currentl using. She looked at her phone bill and saw that one month she had been charged $7.50 for 30 minutes and another month she had been charged $11.25 for 45 minutes of long-distance calling. These calls were made after 6 P.M. to her relatives in Indiana and in Arizona. Draw a graph on the same set of aes ou made for Compan A s figures. Use our graph and what ou know about linear inequalities to advise Rafaella about which compan is best. f. g. h. Getting Read for Section 7.5 (Section 1.5) Evaluate each epression for the given variable value. (a) 2 1 ( 2) (b) 2 1 ( 2) (c) 3 2 ( 1) (d) 3 2 ( 1) (e) 2 2 ( 2) (f) 2 2 ( 2) (g) 2 5 ( 1) (h) 2 5 ( 1) Answers

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14 a. 5 b. 3 c. 1 d. 5 e. 2 f. 2 g. 6 h