Graphing a System of Linear Inequalities. To graph a system of linear inequalities, follow these steps:


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1 3.3 Graph Sstems of Linear Inequalities Before You graphed linear inequalities. Now You will graph sstems of linear inequalities. Wh? So ou can model heart rates during eercise, as in E. 39. Ke Vocabular sstem of linear inequalities solution of a sstem of inequalities graph of a sstem of inequalities The following is an eample of a sstem of linear inequalities in two variables. 8 Inequalit 4 2 > 6 Inequalit 2 A solution of a sstem of inequalities is an ordered pair that is a solution of each inequalit in the sstem. For eample, (5, 22) is a solution of the sstem above. The graph of a sstem of inequalities is the graph of all solutions of the sstem. KEY CONCEPT For Your Notebook Graphing a Sstem of Linear Inequalities To graph a sstem of linear inequalities, follow these steps: STEP Graph each inequalit in the sstem. You ma want to use colored pencils to distinguish the different halfplanes. Identif the region that is common to all the graphs of the inequalities. This region is the graph of the sstem. If ou used colored pencils, the graph of the sstem is the region that has been shaded with ever color. E XAMPLE Graph a sstem of two inequalities REVIEW INEQUALITIES For help with graphing linear inequalities in two variables, see p. 32. Graph the sstem of inequalities. > Inequalit 3 Inequalit 2 STEP Graph each inequalit in the sstem. Use red for > and blue for 3. Identif the region that is common to both graphs. It is the region that is shaded purple. The graph of the sstem is the intersection of the red and blue regions. at classzone.com 68 Chapter 3 Linear Sstems and Matrices
2 E XAMPLE 2 Graph a sstem with no solution Graph the sstem of inequalities. 2 3 < 6 Inequalit 2 2 } 3 4 Inequalit 2 STEP Graph each inequalit in the sstem. Use red for 2 3 < 6 and blue for 2 2 } 3 4. The red and blue regions do not intersect. Identif the region that is common to both graphs. There is no region shaded both red and blue. So, the sstem has no solution. E XAMPLE 3 Graph a sstem with an absolute value inequalit Graph the sstem of inequalities. 3 Inequalit > 4 Inequalit 2 REVIEW ABSOLUTE VALUE For help with graphing absolute value inequalities, see p. 32. STEP Graph each inequalit in the sstem. Use red for 3 and blue for > 4. Identif the region that is common to both graphs. It is the region that is shaded purple. 2 The graph of the sstem is the intersection of the red and blue regions. GUIDED PRACTICE for Eamples, 2, and 3 Graph the sstem of inequalities } > 23 > < > < 3.3 Graph Sstems of Linear Inequalities 69
3 SYSTEMS OF THREE OR MORE INEQUALITIES You can also graph a sstem of three or more linear inequalities, as shown in Eample 4. E XAMPLE 4 Solve a multistep problem SHOPPING A discount shoe store is having a sale, as described in the advertisement shown. Use the information in the ad to write a sstem of inequalities for the regular footwear prices and possible sale prices. Graph the sstem of inequalities. Use the graph to estimate the range of possible sale prices for footwear that is regularl priced at $70. STEP Write a sstem of inequalities. Let be the regular footwear price and let be the sale price. From the information in the ad, ou can write the following four inequalities. 20 Regular price must be at least $ Regular price can be at most $ Sale price is at least ( )% 5 40% of regular price. Sale price is at most (00 2 0)% 5 90% of regular price. STEP 3 Graph each inequalit in the sstem. Then identif the region that is common to all the graphs. It is the region that is shaded. Identif the range of possible sale prices for $70 footwear. From the graph ou can see that when 5 70, the value of is between these values: 0.4(70) 5 28 and 0.9(70) 5 63 So, the value of satisfies c Therefore, footwear regularl priced at $70 sells for between $28 and $63, inclusive, during the sale. Sale price (dollars) Regular price (dollars) GUIDED PRACTICE for Eample 4 7. WHAT IF? In Eample 4, suppose the advertisement showed a range of discounts of 20% 50% and a range of regular prices of $40 $00. a. Write and graph a sstem of inequalities for the regular footwear prices and possible sale prices. b. Use the graph to estimate the range of possible sale prices for footwear that is regularl priced at $ Chapter 3 Linear Sstems and Matrices
4 3.3 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKEDOUT SOLUTIONS on p. WS for Es. 9, 9, and 37 5 STANDARDIZED TEST PRACTICE Es. 2, 3, 26, 27, 36, and 39 5 MULTIPLE REPRESENTATIONS E. 37. VOCABULARY What must be true in order for an ordered pair to be a solution of a sstem of linear inequalities? 2. WRITING Describe how to graph a sstem of linear inequalities. EXAMPLES, 2, and 3 on pp for Es MULTIPLE CHOICE Which sstem of inequalities is represented b the graph? A > 3 B < 24 3 C 22 > 24 D 2 > 24 2 < 3 < 3 SYSTEMS OF TWO INEQUALITIES Graph the sstem of inequalities. 4. > < < < > 4 > > < < < > > > ERROR ANALYSIS Describe and correct the error in graphing the sstem of inequalities EXAMPLE 4 on p. 70 for Es SYSTEMS OF THREE OR MORE INEQUALITIES Graph the sstem of inequalities. 7. < > 26 > > 22 < < < < > 0 23 < 2 3 < > < < > 3 > < < 6 < 4 2 > > Graph Sstems of Linear Inequalities 7
5 26. MULTIPLE CHOICE Which quadrant of the coordinate plane contains no solutions of the sstem of inequalities? A Quadrant I B Quadrant II C Quadrant III D Quadrant IV 27. OPENENDED MATH Write a sstem of two linear inequalities that has (2, 2) as a solution. ABSOLUTE VALUE SYSTEMS Graph the sstem of inequalities. 28. < > > CHALLENGE Write a sstem of linear inequalities for the shaded region. 3. (23, 3) (4, 3) 32. (22, 2) (2, 4) 33. (, 3) (23, 22) (4, 22) (22, 22) (2, 24) (22, 23) (2, 22) PROBLEM SOLVING EXAMPLE 4 on p. 70 for Es SUMMER JOBS You can work at most 20 hours net week. You need to earn at least $92 to cover our weekl epenses. Your dogwalking job pas $7.50 per hour and our job as a car wash attendant pas $6 per hour. Write a sstem of linear inequalities to model the situation. 35. VIDEO GAME SALE An online media store is having a sale, as described in the ad shown. Use the information in the ad to write and graph a sstem of inequalities for the regular video game prices and possible sale prices. Then use the graph to estimate the range of possible sale prices for games that are regularl priced at $ SHORT RESPONSE A book on the care of tropical fish states that the ph level of the water should be between 8.0 and 8.3 ph units and the temperature of the water should be between 768F and 808F. Let be the ph level and be the temperature. Write and graph a sstem of inequalities that describes the proper ph level and temperature of the water. Compare this graph to the graph ou would obtain if the temperatures were given in degrees Celsius. 5 WORKEDOUT SOLUTIONS 72 Chapter 3 Linear on p. Sstems WS and Matrices 5 STANDARDIZED TEST PRACTICE 5 MULTIPLE REPRESENTATIONS
6 37. MULTIPLE REPRESENTATIONS The JuniorSenior Prom Committee must consist of 5 to 8 representatives from the junior and senior classes. The committee must include at least 2 juniors and at least 2 seniors. Let be the number of juniors and be the number of seniors. a. Writing a Sstem Write a sstem of inequalities to describe the situation. b. Graphing a Sstem Graph the sstem ou wrote in part (a). c. Finding s Give two possible solutions for the numbers of juniors and seniors on the prom committee. 38. BASEBALL In baseball, the strike zone is a rectangle the width of home plate that etends from the batter s knees to a point halfwa between the shoulders S and the top T of the uniform pants. The width of home plate is 7 inches. Suppose a batter s knees are 20 inches above the ground and the point halfwa between his shoulders and the top of his pants is 42 inches above the ground. Write and graph a sstem of inequalities that represents the strike zone. S 42 in. T 20 in. 7 in. 39. EXTENDED RESPONSE A person s theoretical maimum heart rate (in heartbeats per minute) is where is the person s age in ears (20 65). When a person eercises, it is recommended that the person strive for a heart rate that is at least 50% of the maimum and at most 75% of the maimum. a. Write a sstem of linear inequalities that describes the given information. b. Graph the sstem ou wrote in part (a). c. A 40earold person has a heart rate of 58 heartbeats per minute when eercising. Is the person s heart rate in the target zone? Eplain. 40. CHALLENGE You and a friend are tring to guess the number of pennies in a jar. You both agree that the jar contains at least 500 pennies. You guess that there are pennies, and our friend guesses that there are pennies. The actual number of pennies in the jar is 000. Write and graph a sstem of inequalities describing the values of and for which our guess is closer than our friend s guess to the actual number of pennies. MIXED REVIEW Evaluate the epression for the given values of and. (p. 0) when 5 4 and when 5 4 and when 5 22 and when 5 5 and 5 26 Solve the inequalit. Then graph the solution. (p. 4) > PREVIEW Prepare for Lesson 3.4 in Es Solve the sstem of linear equations. (p. 60) EXTRA PRACTICE for Lesson 3.3, p ONLINE Graph Sstems QUIZ at of classzone.com Linear Inequalities 73
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