LESSON 4.3 GRAPHING INEQUALITIES

Transcription

1 LESSON.3 GRAPHING INEQUALITIES LESSON.3 GRAPHING INEQUALITIES 9

2 OVERVIEW Here s what ou ll learn in this lesson: Linear Inequalities a. Ordered pairs as solutions of linear inequalities b. Graphing linear inequalities Suppose ou want to know if ou will get an A in our math course, despite a low score on our final. Or ou want to know if our car will get ou to work without running out of gas. Both of these questions can be described b a linear inequalit in two variables. For eample, in the first question ou could ask ourself, What are the possible final scores that will ensure an A in this course, given m overall scores so far? The second question is reall asking, How much gas should I have in the tank to guarantee that m car can travel the entire distance to work? When ou think of the questions like this ou can see that there is going to be more than one possible answer. In fact, ou can see that an value greater than or equal to some minimum score (or amount) will work! Graphing linear inequalities can help ou answer these questions. 70 TOPIC GRAPHING LINEAR EQUATIONS AND INEQUALITIES

3 EXPLAIN LINEAR INEQUALITIES Summar Graphing Linear Inequalities When ou graph a linear equation, all of the points whose coordinates make the equation true lie on a line. For eample, the linear equation + = 3 is plotted in Figure.3.. Notice that this line divides the -plane into three regions: points above the line points on the line points below the line You know that the coordinates of an point on the line satisf the equation + = 3. Using inequalit smbols ou can also describe which points lie above the line and which points lie below the line. For eample, tr substituting the coordinates of some points not on the line + = 3 into the inequalities below to see which give ou true statements: Figure.3. + = 3 + < 3 + > 3 (0, 0) Is < 3? Is > 3? Is 0 < 3? YES Is 0 > 3? NO (, ) Is + < 3? Is + > 3? Is 8 < 3? NO Is 8 > 3? YES (, ) Is + < 3? Is + > 3? Is 3 < 3? YES Is 3 > 3? NO (3, 5) Is < 3? Is > 3? Is 8 < 3? NO Is 8 > 3? YES (, 3) Is + ( 3) < 3? Is + ( 3) > 3? Is < 3? YES Is > 3? NO (, 7) Is ( ) + 7 < 3? Is ( ) + 7 > 3? Is 5 < 3? NO Is 5 > 3? YES (, ) Is ()+ () < 3? Is () + () > 3? Is 0 < 3? YES Is 0 > 3? NO These points are plotted in Figure.3.. Notice that the coordinates of an single point can make onl one of the inequalities true. The points whose coordinates make the inequalit + < 3 true lie below the line; the points whose coordinates make the inequalit + > 3 true lie above the line. (, 7) (3, 5) (, ) (, ) (0, 0) (, 3) + = 3 (, ) Figure.3. The coordinates of an given point can make onl one of these inequalities true since, in this eample, a number cannot be both greater than 3 and less than 3. LESSON.3 GRAPHING INEQUALITIES EXPLAIN 7

4 If the coordinates of an point above the corresponding line make an inequalit true, then the coordinates of all the points above the line make the inequalit true. Similarl, if the coordinates of an point below the corresponding line make the inequalit true, then the coordinates of all the points below the line make the inequalit true. You can use this fact to figure out in which region the points that satisf an inequalit lie. Wh can t ou use a test point that s on the line? Because, if ou do, ou still won t know which region to shade. If (0, 0) doesn't lie on the line, it's a good test point to choose because when ou substitute it into the inequalit, the math is eas. A dotted line is used if the inequalit is < or > to show that points on the line do not satisf the inequalit. To find the region whose points satisf an inequalit, pick a point that does not lie on the corresponding line and substitute its coordinates into the inequalit. If the coordinates of the point make the inequalit true, then shade the region in which the point lies. If the coordinates of the point do not make the inequalit true, then shade the region in which the point does not lie. To graph a linear inequalit:. Graph the linear equation that corresponds to the given inequalit. Use a dotted line.. Use a test point that is not on the line to determine the region whose points satisf the inequalit. = 3. Shade that region. If the inequalit is or, shade the line as well (that is, make it a solid line). For eample, to graph the linear inequalit :. Graph the corresponding equation =. Use a dotted line. See Figure Pick a point not on the line, sa (0, 0), and substitute it into the inequalit. Figure.3.3 Is 0 0? Is 0? No. ; ; ;; Figure Since this statement isn't true, the point (0, 0) does not satisf the inequalit. So, shade the region in which (0, 0) does not lie. Shade the line because points on the line also satisf the inequalit. See Figure TOPIC GRAPHING LINEAR EQUATIONS AND INEQUALITIES

5 Sample Problems Answers to Sample Problems. Graph the inequalit < 3. a. Graph the corresponding equation = 3. = 3 c. < 3 b. Substitute a test point into the < 3 inequalit, sa (0, 0). Is ( ) < 3? Is < 3? b. 0, 0 0, Yes c. Shade the region whose points satisf the inequalit.. Graph the inequalit 3 0. a. Graph the corresponding equation 3 = 0. a., c. 3 0 b. Substitute a test point into the inequalit, sa (, ). (You can't substitute (0, 0) since it lies on the line 3 = 0.) 3 0 Is 3( ) 0? Is 0? b.,, No c. Shade the region whose points satisf the inequalit. LESSON.3 GRAPHING INEQUALITIES EXPLAIN 73

6 HOMEWORK Homework Problems Circle the homework problems assigned to ou b the computer, then complete them below. Eplain Linear Inequalities. In each row, check the boes corresponding to the coordinates that make the statement true. < + = + > + (, 7) (, ) (, 5) (0, 0) (, 3) (, 3) (3, ). Graph the inequalit Graph the inequalit Find the coordinates of three points that satisf the inequalit Graph the inequalit. 7. In what was do the graphs of the inequalities + < and + differ? 8. Graph the inequalit > Janna has up to $5.00 to spend on snacks at a new health food store. If guava chips are $.00 per pound and shredded coconut is $.50 per pound, graph the inequalit that represents how much of each she can bu. 0. Shobana has up to $.00 to spend on junk food. If she can bu bulk cand for $.5 per pound and cookies for $3.00 per pound, graph the inequalit that represents how much of each she can bu.. Graph the inequalit In what was do the graphs of the inequalities > 3 and < 3 differ?. Graph the inequalit 3 >. 7 TOPIC GRAPHING LINEAR EQUATIONS AND INEQUALITIES

7 APPLY Practice Problems Here are some additional practice problems for ou to tr.. In each row, check the boes corresponding to the coordinates that make the statement true. (, ) (, ) ( 5, ) (3, 8) ( 3, ) (, 3) (, ) < = >. In each row, check the boes corresponding to the coordinates that make the statement true. (, 3) (3, ) (, ) (, 7) (, ) (, 5) (, 5) + < + = + > 3. In each row, check the boes corresponding to the coordinates that make the statement true. ( 3, ) ( 5, 3) (, ) (3, 5) (3, 8) (5, ) (, 5) < = >. In each row, check the boes corresponding to the coordinates that make the statement true. (, 7) (, ) (, ) (, 5) (, 5) (, ) (, ) 3 + < 3 + = 3 + > 5. Graph the inequalit >.. Graph the inequalit < Graph the inequalit < Graph the inequalit + >. 9. Graph the inequalit + < Graph the inequalit + <.. Graph the inequalit +.. Graph the inequalit Graph the inequalit +.. Graph the inequalit. 5. Graph the inequalit.. Graph the inequalit Graph the inequalit + < Graph the inequalit + >. 9. Graph the inequalit + >. 0. Graph the inequalit 3.. Graph the inequalit 3.. Graph the inequalit. 3. Graph the inequalit 3 <.. Graph the inequalit + 5 > Graph the inequalit 3 + > 8.. Graph the inequalit Graph the inequalit Graph the inequalit. LESSON.3 GRAPHING INEQUALITIES APPLY 75

8 Practice Test EVALUATE Take this practice test to be sure that ou are prepared for the final quiz in Evaluate. 3. Graph the inequalit >.. The graph of the line + = is shown in Figure.3.5. Circle the point(s) below that satisf the inequalit +. + = (0, 0) (5, ) ( 3, ) (8, ) Figure Graph the inequalit.. Circle the point(s) below that satisf the linear inequalit. (3, 5) (0, 0) (8, ) (, 7) 5. Circle the inequalit below that has a solution represented on the graph shown in Figure > 0 5 < > 0 5 < 0. Graph the inequalit The graph of the equation = is shown in Figure.3.7. Circle the point(s) below that satisf the inequalit >. (, ) ( 3, 3) ( 5, ) (, 5) Figure Circle the inequalit below that has a solution represented on the graph shown in Figure.3.8. > + Figure.3.8 = < Figure.3. 7 TOPIC GRAPHING LINEAR EQUATIONS AND INEQUALITIES

9 TOPIC CUMULATIVE ACTIVITIES CUMULATIVE REVIEW PROBLEMS These problems combine all of the material ou have covered so far in this course. You ma want to test our understanding of this material before ou move on to the net topic. Or ou ma wish to do these problems to review for a test.. Graph the equation =.. Find the - and -intercepts of the line + = Find the equation of the line through the point (9, ) with 5 slope. Write the equation in standard form.. Graph the line through the point (3, 3) with slope. 5. Evaluate the epression 5a b a 3 + 7b when a = and b =.. Find: [3 + (5 7) ] Graph the inequalit 3. 8a. Find the equation of the vertical line through the point (, 5). 8b. What is the slope of this line? 8c. Find the equation of the horizontal line through the point (, 5). 8d. What is the slope of this line? 9. Write in lowest terms: 0. Graph the line =.. Find: The sum of three consecutive numbers is 8. What are the numbers? 3. Graph the inequalit +.. Find the - and -intercepts of the line =. 5a. Find four points whose coordinates satisf the inequalit b. Find four points whose coordinates satisf the inequalit c. Find four points whose coordinates satisf the equation + = 5.. Graph the equation 3 + =. 7. Find: 8 8. Solve for : = ( 3 + ) 8 9. Find: Graph the line through the point (, 5) with slope.. Find the equation of the line through the points ( 7, 3) and (0, ). Write our answer in standard form.. Write in lowest terms: 3. Solve for : 5( ) = ( ). Graph the inequalit. 5. Paul is ears older than Rita. Ten ears ago, Paul was twice as old as Rita was then. How old is each of them now?. The surface area, S, of a sphere is S = r, where r is the radius of the sphere. Solve this formula for r. 7. Graph the line = Evaluate the epression when = 3 and = Solve for, then graph its solution on the number line below Find the slope of the line through the points (3, 8) and (, ). TOPIC CUMULATIVE REVIEW 77

10 3. Graph the inequalit Write in lowest terms: 33. Solve for : + = (7 + 9 ) 3. Write the equation of the line through the point ( 8, ) with 9 slope. Write our answer in slope-intercept form Find the slope of the line perpendicular to the line through the points (7, 3) and (, 9). 3. Graph the equation + =. 37. Find the equation of the line through the point (5, ) with slope. Write our answer in point-slope form. 38. Find the slope and -intercept of the line 3 + = Find: Solve 7 for, then graph its solution on the number line below TOPIC GRAPHING LINEAR EQUATIONS AND INEQUALITIES