L-13 Graph Linear Inequalities in Two Variables Objective: TLW Graph Linear Inequalities with two variables
The x-intercept of a line is the point at which the line crosses the x axis. ( i.e. where the y value = 0 ) x-intercept = ( x, 0 ) The y-intercept of a line is the point at which the line crosses the y axis. ( i.e. where the x value = 0 ) y-intercept = ( 0, y )
EXAMPLE 1 Standardized Test Practice Ordered pair Substitute Conclusion (6, 3) 3(6) + 4( 3) = 6 > 8 (6, 3 ) is not a solution (0, 2) 3(0) + 4(2) = 8 > 8 (0, 2 ) is not a solution ( 2, 1) ( 3, 5) 3( 2) + 4( 1) = 10 > 8 3( 3) + 4(5) = 11 > 8 ( 2, 1) is not a solution ( 3, 5) is a solution
GUIDED PRACTICE for Example 1 Tell whether the given ordered pair is a solution of 5x 2y 6. Ordered pair Substitute Conclusion 1. (0, 4) 5(0) 2( 4) = 8 < 6 (0, 4 ) is not a solution 2. (2, 2) 5(2) 2(2) = 6 < 6 (2, 2 ) is a solution 3. ( 3, 8) 5( 3) 2(8) = 31 < 6 ( 3, 8 ) is a solution 4. ( 1, 7) ( 1, 7 ) is 5( 1) 2( 7) = 9 < 6 not a solution
To graph a linear inequality in two variables (say, x and y), first get y alone on one side. Then consider the related equation obtained by changing the inequality sign to an equals sign. 6x + 2y 6 6x + 2y = 6 y =-3x+ 3 y -3x+3
EXAMPLE 2 Graph linear inequalities with one variable Graph (a) y < 3 & (b) x < 2 in a coordinate plane. a. Graph the boundary line y = 3. Use a solid line because the inequality symbol is <. Test the point (0,0). Because (0,0) is not a solution of the inequality, shade the halfplane that does not contain (0,0).
EXAMPLE 2 Graph linear inequalities with one variable b. Graph the boundary line x = 2.Use a dashed line because the inequality symbol is <. Test the point (0,0). Because (0,0) is a solution of the inequality, shade the halfplane that does not contains (0,0).
EXAMPLE 3 Graph linear inequalities with two variables Graph (a) y > 2x and (b) 5x 2y 4 in a coordinate plane. a. Graph the boundary line y = 2x. Use a dashed line because the inequality symbol is >. Test the point (1,1). Because (1,1) is a solution of the inequality, shade the halfplane that contains (1,1).
EXAMPLE 3 Graph linear inequalities with two variables b. Graph the boundary line 5x 2y = 4.Use a solid line because the inequality symbol is <. Test the point (0,0). Because (0,0) is not a solution of the inequality, shade the half-plane that does not contain (0,0).
GUIDED PRACTICE for Examples 2 and 3 Graph the inequality in a coordinate plane. 5. y > 1 SOLUTION Graph the boundary line y = 1.Use a solid line because the inequality symbol is >. Test the point (0,0). Because (0,0) is a solution of the inequality, shade the half- plane that contains (0,0).
GUIDED PRACTICE for Examples 2 and 3
GUIDED PRACTICE for Examples 2 and 3 Graph the inequality in a coordinate plane. 6. x > 4 SOLUTION Graph the boundary line x = 4.Use a dashed line because the inequality symbol is >. Test the point (0,0). Because (0,0) is a solution of the inequality, shade the halfplane that contains (0,0).
GUIDED PRACTICE for Examples 2 and 3
GUIDED PRACTICE for Examples 2 and 3 Graph the inequality in a coordinate plane. 7. y > 3x SOLUTION Graph the boundary line y = 3x.Use a dashed line because the inequality symbol is >. Test the point (1,1). Because (1,1) is a solution of the inequality, shade the halfplane that contains (1,1).
GUIDED PRACTICE for Examples 2 and 3
GUIDED PRACTICE for Examples 2 and 3 Graph the inequality in a coordinate plane. 8. y < 2x +3 SOLUTION Graph the boundary line y = 2x + 3.Use a solid line because the inequality symbol is <. Test the point (0,0). Because (0,0) is a solution of the inequality, shade the half-plane that contains (0,0).
GUIDED PRACTICE for Examples 2 and 3
GUIDED PRACTICE for Examples 2 and 3 Graph the inequality in a coordinate plane. 9. x + 3y < 9 SOLUTION Graph the boundary line x + 2y = 9.Use a solid line because the inequality symbol is <. Test the point (0,0). Because (0,0) is a solution of the inequality, shade the half-plane that contains (0,0).
GUIDED PRACTICE for Examples 2 and 3
GUIDED PRACTICE for Examples 2 and 3 Graph the inequality in a coordinate plane. 10. 2x 6y > 9 SOLUTION Graph the boundary line 2x 6y = 9.Use a solid line because the inequality symbol is >. Test the point (0,0). Because (0,0) is not a solution of the inequality, shade the halfplane that does not contains (0,0).
GUIDED PRACTICE for Examples 2 and 3
EXAMPLE 4 Solve a multi-step problem Movie Recording A film class is recording a DVD of student-made short films. Each student group is allotted up to 300 megabytes (MB) of video space. The films are encoded on the DVD at two different rates: a standard rate of 0.4 MB/sec for normal scenes and a high-quality rate of 1.2 MB/sec for complex scenes.
EXAMPLE 4 Solve a multi-step problem Write an inequality describing the possible amounts of time available for standard and high-quality video. Graph the inequality. Identify three possible solutions of the inequality.
EXAMPLE 4 Solve a multi-step problem STEP 1 Write an inequality. First write a verbal model. An inequality is 0.4x + 1.2y 300
EXAMPLE 4 Solve a multi-step problem STEP 2 Graph the inequality. First graph the boundary line 0.4x + 1.2y = 300. Use a solid line because the inequality symbol is. Test the point (0, 0). Because (0, 0) is a solution of the inequality, shade the half-plane that contains (0, 0). Because x and y cannot be negative, shade only points in the first quadrant.
EXAMPLE 4 Solve a multi-step problem STEP 3 Identify solutions. Three solutions are given below and on the graph. (150,200) 150 seconds of standard and 200 seconds of high quality (300, 120) 300 seconds of standard and 120 seconds of high quality (600, 25) 600 seconds of standard and 25 seconds of high quality For the first solution, 0.4(150) + 1.2(200) = 300, so all of the available space is used. For the other two solutions, not all of the space is used.
GUIDED PRACTICE for Examples 4 and 5 STEP 3 Identify solutions. Three solutions are given below and on the graph. (300,200) 300 seconds of standard and 200 seconds of high quality (600, 150) 600 seconds of standard and 150 seconds of high quality (100, 300) 100 seconds of standard and 300 seconds of high quality For the second solution, 0.4(600) + 1.2(150) = 420, so all of the available space is used. For the other two solutions, not all of the space is used.