Lesson 5. Eercises, pages 6 68 A. Determine whether each point is a solution of the given inequalit. a) - -16 A(-, ) In the inequalit, substitute:, L.S.: ( ) () 17 R.S. 16 Since the L.S.<R.S., the point is not a solution. b) - 5 B(-1, 1) In the inequalit, substitute: 1, 1 L.S.: ( 1) 1 5 R.S. 5 Since the L.S.<R.S., the point is a solution. c) 7-7 C(-, -5) In the inequalit, substitute:, 5 L.S.: ( 5) 15 R.S.: ( ) 7 11 Since the L.S.<R.S., the point is not a solution. d) 5 - + 8 < D(6, 7) In the inequalit, substitute: 6, 7 L.S.: 5(6) (7) 8 R.S. Since the L.S.>R.S., the point is not a solution.. Match each graph with an inequalit below. i) + - ii) + > iii) - < iv) - -1 a) b) The line has slope and -intercept, so its equation is:, or The inequalit is: > The line has slope.5 and -intercept.5, so its equation is:.5.5, or 1 The inequalit is:» 1 8 5. Graphing Linear Inequalities in Two Variables Solutions DO NOT COPY. P
c) d) The line has slope and -intercept, so its equation is:, or The inequalit is: The line has slope.5 and -intercept 1, so its equation is:.5 1, or The inequalit is: < 5. Write an inequalit to describe each graph. a) b) The equation can be The equation can be written as: written as: The line is broken, and The line is solid, and the shaded region is the shaded region is below the line so an below the line so an inequalit is: <, inequalit is:, or < or» c) d) The equation can be The equation can be written as: written as: The line is broken, and The line is broken, and the shaded region is the shaded region is above the line so an above the line so the inequalit is: >, inequalit is: >, or < or > P DO NOT COPY. 5. Graphing Linear Inequalities in Two Variables Solutions 9
B 6. Graph each linear inequalit. a) + 5 b) 7-1 + 1 Graph of 5 5 1 Graph of 1 1 1 Use intercepts to graph the related functions. When, 5 When, 1 When,.5 When, Draw a solid line. Shade Draw a broken line. Shade the region below the line. c) 6 - - d) - Graph of 6 Graph of 6 Use intercepts to graph the related functions. When, When, When, 1 When, 1.5 Draw a broken line. Shade Draw a solid line. Shade the region below the line. 1 5. Graphing Linear Inequalities in Two Variables Solutions DO NOT COPY. P
7. Graph each linear inequalit. Give the coordinates of points that satisf the inequalit. a) 5 + 7 15 b) - -9 Graph of 5 15 Graph of 9 5 15 6 9 Use intercepts to graph the related functions. When, 5 When,.5 When, When, L.S. ; R.S. 15 L.S. ; R.S. 9 Since <15, the origin Since > 9, the origin does not lie in the shaded region. does not lie in the shaded region. Draw a broken line. Shade Draw a solid line. Shade From the graph, points From the graph, points that satisf the inequalit that satisf the inequalit are: (, ), (1, 5), (, ) are: (, ), ( 1, ), ( 1, 6) c) + 6 - d) - 7 6 1 Graph of 6 6 Graph of 7 1 6 6 7 1 Graph the related functions. When, When, When 1, When 1, 7 L.S. ; R.S. L.S. ; R.S. 1 Since >, the origin Since <1, the origin lies in the shaded region. lies in the shaded region. Draw a solid line. Shade Draw a broken line. Shade From the graph, points From the graph, points that satisf the inequalit that satisf the inequalit are: (, 1), (1, ), (, ) are: ( 1, ), (1, 1), (, ) P DO NOT COPY. 5. Graphing Linear Inequalities in Two Variables Solutions 11
8. Write an inequalit to describe each graph. a) b) 6 The line has slope and The line has slope and -intercept 1, so its -intercept 5, so its equation is: 1 equation is: 5, The line is solid and the The line is broken and the region below is shaded. region below is shaded. An inequalit is: An inequalit is: 1 < 5 9. A student graphed the inequalit - 6 and used the origin as a test point. Could the student then shade the correct region of the graph? Eplain our answer. No, the line passes through the origin, so it cannot be used as a test point. The test point must not lie on the line that divides the region. 1. Use technolog to graph each linear inequalit. Sketch the graph. a) < 1.6-1.95 b) > 9 + 7 Graph: 1.6 1.95 Graph: 9 7 The boundar is not part The boundar is not part of the graph. of the graph. c) 8 - - 5 d).8 +. -.7 8 5 8 5 Graph: 8 5 The boundar is part of the graph...8.7.8.7.. Graph:.8.7.. The boundar is part of the graph. 1 5. Graphing Linear Inequalities in Two Variables Solutions DO NOT COPY. P
11. Nina takes her friends to an ice cream store. A milkshake costs $ and a chocolate sundae costs $.5. Nina has $18 in her purse. a) Write an inequalit to describe how Nina can spend her mone. Let m represent the number of milkshakes and s represent the number of sundaes. An inequalit is: m.5s 18 b) Determine possible was Nina can spend up to $18. Determine the coordinates of points that satisf the related function. When s, m 6 When s 6, m 1 Join the points with a solid line. The solution is the points, with whole-number coordinates, on and below the line. Three was are: milkshakes, sundaes; milkshakes, sundaes; milkshakes, sundaes 6 s m.5s 18 m 6 c) What is the most mone Nina can spend and still have change from $18? The point, with whole-number coordinates, that is closest to the line has coordinates (5, 1); the cost, in dollars, is: (5)() 1(.5) 17.5 Nina can spend $17.5 and still have change. 1. The relationship between two negative numbers p and q is described b the inequalit p - q 7-6. a) What are the restrictions on the variables? Since the numbers are negative, p< and q< b) Graph the inequalit. Determine the coordinates of points that satisf the related function. When p 1, q When p 6, q Draw a broken line through the points. The solution is the points below the line in Quadrant. Graph of p q 6 q 1 8 1 p c) Write the coordinates of points that satisf the inequalit. Sample response: Two points are: (, ) and ( 1, ) P DO NOT COPY. 5. Graphing Linear Inequalities in Two Variables Solutions 1
1. Graph each inequalit for the given restrictions on the variables. a) > - + ; for >, > Since >, >, the graph is in Quadrant 1. The graph of the related function has slope and -intercept. Draw a broken line to represent the related function in Quadrant 1. Shade The aes bounding the graph are broken lines. b) - < 6; for, Since»,, the graph is in Quadrant. Graph the related function. When, When, Draw a broken line in Quadrant. L.S. ; R.S. 6 Since <6, the origin lies in the shaded region. Shade Graph of,, 6 6 Graph of 6,, 6 c) + 5 - > ; for, Since,», the graph is in Quadrant. Graph the related function. When, When 5, 8 Draw a broken line in Quadrant. L.S. ; R.S. Since <, the origin does not lie in the shaded region. Shade Graph of 5,, 6 6 1. a) For A(9, a) to be a solution of - < 5, what must be true about a? Substitute the coordinates of A in the inequalit. (9) (a)<5 Solve for a. a> a>11 b) For B(b, -) to be a solution of + -1, what must be true about b? Substitute the coordinates of B in the inequalit. (b) ( )» 1 Solve for b. b» b» 1 5. Graphing Linear Inequalities in Two Variables Solutions DO NOT COPY. P
15. A personal trainer books clients for either 5-min or 6-min appointments. He meets with clients a maimum of h each week. a) Write an inequalit that represents the trainer s weekl appointments. Let represent the number of 5-min appointments and represent the number of 6-min appointments. An inequalit is: 5 6 Divide b 15. 16 b) Graph the related equation, then describe the graph of the inequalit. Determine the coordinates of points that satisf the related function. When, When, 5 Join the points with a solid line. The solution is the points, with whole-number coordinates, on and below the line. 5 1 16 1 5 6 c) How man 5-min appointments are possible if no 6-min appointments are scheduled? Where is the point that represents this situation located on the graph? For no 6-min appointments,, so the point is on the -ais; it is the point with whole-number coordinates that is closest to the -intercept of the graph of the related equation. When,, or 5. 5 Fift-three 5-min appointments are possible. 16. Graph this inequalit. Identif the strateg ou used and eplain wh ou chose that strateg. + 1 Graph the related function. Determine the intercepts. When, When, Draw a solid line. L.S. ; R.S. 1 Since <1, the origin is not in the shaded region. Shade Graph of 1 P DO NOT COPY. 5. Graphing Linear Inequalities in Two Variables Solutions 15
C 17. How is a linear inequalit in two variables similar to a linear inequalit in one variable? How are the inequalities different? The solutions of both inequalities are usuall sets of values. A linear inequalit in one variable is a set of numbers that can be represented on a number line. A linear inequalit in two variables is a set of ordered pairs that can be represented on a coordinate plane. 16 5. Graphing Linear Inequalities in Two Variables Solutions DO NOT COPY. P