Essential Question How can you graph a linear inequality in two variables?

Transcription

1 5.6 Graphing Linear Inequalities in Two Variables Essential Question How can ou graph a linear inequalit in two variables? A solution of a linear inequalit in two variables is an ordered pair (, ) that makes the inequalit true. The graph of a linear inequalit in two variables shows all the solutions of the inequalit in a coordinate plane. Writing a Linear Inequalit in Two Variables Work with a partner. a. Write an equation represented b the dashed line. b. The solutions of an inequalit are represented b the shaded region. In words, describe the solutions of the inequalit. c. Write an inequalit represented b the graph. Which inequalit smbol did ou use? Eplain our reasoning. USING TOOLS STRATEGICALLY To be proficient in math, ou need to use technological tools to eplore and deepen our understanding of concepts. Using a Graphing Calculator Work with a partner. Use a graphing calculator to graph. a. Enter the equation = into our calculator. b. The inequalit has the smbol. So, the region to be shaded is above the graph of =, as shown. Verif this b testing a point in this region, such as (0, 0), to make sure it is a solution of the inequalit Because the inequalit smbol is greater than or equal to, the line is solid and not dashed. Some graphing calculators alwas use a solid line when graphing inequalities. In this case, ou have to determine whether the line should be solid or dashed, based on the inequalit smbol used in the original inequalit. Graphing Linear Inequalities in Two Variables Work with a partner. Graph each linear inequalit in two variables. Eplain our steps. Use a graphing calculator to check our graphs. a. > + 5 b. + c. 5 Communicate Your Answer. How can ou graph a linear inequalit in two variables? 5. Give an eample of a real-life situation that can be modeled using a linear inequalit in two variables. Section 5.6 Graphing Linear Inequalities in Two Variables 67

2 5.6 Lesson What You Will Learn Core Vocabular linear inequalit in two variables, p. 68 solution of a linear inequalit in two variables, p. 68 graph of a linear inequalit, p. 68 half-planes, p. 68 Previous ordered pair Check solutions of linear inequalities. Graph linear inequalities in two variables. Use linear inequalities to solve real-life problems. Linear Inequalities A linear inequalit in two variables, and, can be written as a + b < c a + b c a + b > c a + b c where a, b, and c are real numbers. A solution of a linear inequalit in two variables is an ordered pair (, ) that makes the inequalit true. Checking Solutions Tell whether the ordered pair is a solution of the inequalit. a. + < ; (, 9) b. 8; (, ) a. + < Write the inequalit. ( ) + 9 <? Substitute for and 9 for. 7 < Simplif. 7 is not less than. So, (, 9) is not a solution of the inequalit. b. 8 Write the inequalit. ()? 8 Substitute for and for. 8 8 Simplif. 8 is equal to 8. So, (, ) is a solution of the inequalit. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Tell whether the ordered pair is a solution of the inequalit.. + > 0; (, ). 5; (0, 0). 5 ; (, ). < 5; (5, 7) READING A dashed boundar line means that points on the line are not solutions. A solid boundar line means that points on the line are solutions. Graphing Linear Inequalities in Two Variables The graph of a linear inequalit in two variables shows all the solutions of the inequalit in a coordinate plane. All solutions of < lie on one side of the boundar line =. The boundar line divides the coordinate plane into two half-planes. The shaded half-plane is the graph of <. 68 Chapter 5 Solving Sstems of Linear Equations

3 Core Concept Graphing a Linear Inequalit in Two Variables Step Graph the boundar line for the inequalit. Use a dashed line for < or >. Use a solid line for or. Step Test a point that is not on the boundar line to determine whether it is a solution of the inequalit. Step When the test point is a solution, shade the half-plane that contains the point. When the test point is not a solution, shade the half-plane that does not contain the point. STUDY TIP It is often convenient to use the origin as a test point. However, ou must choose a different test point when the origin is on the boundar line. Graph in a coordinate plane. Graphing a Linear Inequalit in One Variable Step Graph =. Use a solid line because the inequalit smbol is. Step Test (0, 0). Write the inequalit. 0 Substitute. Step Because (0, 0) is a solution, shade the half-plane that contains (0, 0). (0, 0) Graph + > in a coordinate plane. Graphing a Linear Inequalit in Two Variables Step Graph + =, or = +. Use a dashed line because the inequalit smbol is >. Check Step Test (0, 0). 5 + > (0) + (0) >? Write the inequalit. Substitute. (0, 0) 0 > Simplif. Step Because (0, 0) is not a solution, shade the half-plane that does not contain (0, 0). Monitoring Progress Help in English and Spanish at BigIdeasMath.com Graph the inequalit in a coordinate plane. 5. > < 0 Section 5.6 Graphing Linear Inequalities in Two Variables 69

4 Solving Real-Life Problems Modeling with Mathematics You can spend at most $0 on grapes and apples for a fruit salad. Grapes cost $.50 per pound, and apples cost $ per pound. Write and graph an inequalit that represents the amounts of grapes and apples ou can bu. Identif and interpret two solutions of the inequalit.. Understand the Problem You know the most that ou can spend and the prices per pound for grapes and apples. You are asked to write and graph an inequalit and then identif and interpret two solutions.. Make a Plan Use a verbal model to write an inequalit that represents the problem. Then graph the inequalit. Use the graph to identif two solutions. Then interpret the solutions.. Solve the Problem Words Cost per pound of grapes Pounds Cost per of grapes + pound of Pounds Amount of apples ou can apples spend Pounds of apples Check Fruit Salad (, 6) (, 5) Pounds of grapes () + 6? () + 5? Variables Let be pounds of grapes and be pounds of apples. Inequalit Step Graph.5 + = 0, or = Use a solid line because the inequalit smbol is. Restrict the graph to positive values of and because negative values do not make sense in this real-life contet. Step Test (0, 0) Write the inequalit..5(0) + 0? 0 Substitute. 0 0 Simplif. Step Because (0, 0) is a solution, shade the half-plane that contains (0, 0). One possible solution is (, 6) because it lies in the shaded half-plane. Another possible solution is (, 5) because it lies on the solid line. So, ou can bu pound of grapes and 6 pounds of apples, or pounds of grapes and 5 pounds of apples.. Look Back Check our solutions b substituting them into the original inequalit, as shown. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 9. You can spend at most $ on red peppers and tomatoes for salsa. Red peppers cost $ per pound, and tomatoes cost $ per pound. Write and graph an inequalit that represents the amounts of red peppers and tomatoes ou can bu. Identif and interpret two solutions of the inequalit. 70 Chapter 5 Solving Sstems of Linear Equations

5 5.6 Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. VOCABULARY How can ou tell whether an ordered pair is a solution of a linear inequalit?. WRITING Compare the graph of a linear inequalit in two variables with the graph of a linear equation in two variables. Monitoring Progress and Modeling with Mathematics In Eercises 0, tell whether the ordered pair is a solution of the inequalit. (See Eample.). + < 7; (, ). 0; (5, ) 5. + ; ( 9, ) > 6; (, ) ; (, ) 8. 5 ; (, ) 9. 6 > ; ( 8, ) 0. 8 < 5; ( 6, ) In Eercises 6, tell whether the ordered pair is a solution of the inequalit whose graph is shown.. (0, ). (, ). (, ). (0, 0) 5. (, ) 6. (, ) 7. MODELING WITH MATHEMATICS A carpenter has at most $50 to spend on lumber. The inequalit represents the numbers of -b-8 boards and the numbers of -b- boards the carpenter can bu. Can the carpenter bu twelve -b-8 boards and fourteen -b- boards? Eplain. In Eercises 9, graph the inequalit in a coordinate plane. (See Eample.) > 6. <.. > 7. < 9 In Eercises 5 0, graph the inequalit in a coordinate plane. (See Eample.) 5. > < > ERROR ANALYSIS In Eercises and, describe and correct the error in graphing the inequalit.. < +. in. in. 8 ft $ each in. 8 in. 8 ft $8 each 8. MODELING WITH MATHEMATICS The inequalit + 9 represents the numbers of multiplechoice questions and the numbers of matching questions ou can answer correctl to receive an A on a test. You answer 0 multiple-choice questions and 8 matching questions correctl. Do ou receive an A on the test? Eplain. Section 5.6 Graphing Linear Inequalities in Two Variables 7

6 . MODELING WITH MATHEMATICS You have at most $0 to spend at an arcade. Arcade games cost $0.75 each, and snacks cost $.5 each. Write and graph an inequalit that represents the numbers of games ou can pla and snacks ou can bu. Identif and interpret two solutions of the inequalit. (See Eample.). MODELING WITH MATHEMATICS A drama club must sell at least $500 worth of tickets to cover the epenses of producing a pla. Write and graph an inequalit that represents how man adult and student tickets the club must sell. Identif and interpret two solutions of the inequalit. 0. HOW DO YOU SEE IT? Match each inequalit with its graph. a. 6 b. < 6 c. > 6 d. 6 A. C. B. D. In Eercises 5 8, write an inequalit that represents the graph REASONING When graphing a linear inequalit in two variables, wh must ou choose a test point that is not on the boundar line?. THOUGHT PROVOKING Write a linear inequalit in two variables that has the following two properties (0, 0), (0, ), and (0, ) are not solutions. (, ), (, ), and (, ) are solutions. 5. WRITING Can ou alwas use (0, 0) as a test point when graphing an inequalit? Eplain. 9. PROBLEM SOLVING Large boes weigh 75 pounds, and small boes weigh 0 pounds. Weight limit: a. Write and graph 000 lb an inequalit that represents the numbers of large and small boes a 00-pound deliver person can take on the elevator. b. Eplain wh some solutions of the inequalit might not be practical in real life. Maintaining Mathematical Proficienc Write the net three terms of the arithmetic sequence. (Section.6) CRITICAL THINKING In Eercises and 5, write and graph an inequalit whose graph is described b the given information.. The points (, 5) and (, 5) lie on the boundar line. The points (6, 5) and (, ) are solutions of the inequalit. 5. The points ( 7, 6) and (, 8) lie on the boundar line. The points ( 7, 0) and (, ) are not solutions of the inequalit. Reviewing what ou learned in previous grades and lessons 6. 0, 8, 6,,, , 8,,, 7,... 8.,,,, 5,... 7 Chapter 5 Solving Sstems of Linear Equations