2 5 Linear Inequalities in Two Variables Warm

2 -5 Linear Inequalities in Two Variables Warm Up Find the intercepts of each line. 1. 3 x + 2 y = 18 (0, 9), (6, 0) 2. 4 x – y = 8 (0, – 8), (2, 0) 3. 5 x + 10 = 2 y (0, 5), (– 2, 0) Write the function in slope-intercept form. Then graph. 4. 2 x + 3 y = – 3 Holt Algebra 2

Linear. Inequalities in Variables Linear in Two 2 -5 Two Variables Holt Algebra 22

2 -5 Linear Inequalities in Two Variables Objectives Can you graph linear inequalities on the coordinate plane? Can you solve problems using linear inequalities? Holt Algebra 2

2 -5 Linear Inequalities in Two Variables Linear functions form the basis of linear inequalities. A linear inequality in two variables relates two variables using an inequality symbol, such as y > 2 x – 4. Its graph is a region of the coordinate plane bounded by a line. The line is a boundary line, which divides the coordinate plane into two regions. Holt Algebra 2

2 -5 Linear Inequalities in Two Variables To graph y ≥ 2 x – 4, make the boundary line solid, and shade the region above the line. To graph y > 2 x – 4, make the boundary line dashed because y-values equal to 2 x – 4 are not included. Holt Algebra 2

2 -5 Linear Inequalities in Two Variables Example 1 Graph the inequality The boundary line is y-intercept of 2 and a slope of Draw the boundary line dashed because it is not part of the solution. Then shade the region above the boundary line to show. Holt Algebra 2 . which has a.

2 -5 Linear Inequalities in Two Variables Example 2 Graph the inequality y ≤ – 1. Recall that y= – 1 is a horizontal line. Step 1 Draw a solid line for y=– 1 because the boundary line is part of the graph. Step 2 Shade the region below the boundary line to show where y < – 1. Holt Algebra 2

2 -5 Linear Inequalities in Two Variables Example 3 Graph the inequality y ≥ 3 x – 2. The boundary line is y = 3 x – 2 which has a y–intercept of – 2 and a slope of 3. Draw a solid line because it is part of the solution. Then shade the region above the boundary line to show y > 3 x – 2. Holt Algebra 2

2 -5 Linear Inequalities in Two Variables Example 4 Graph the inequality y < – 3. Recall that y = – 3 is a horizontal line. Step 1 Draw the boundary line dashed because it is not part of the solution. Step 2 Shade the region below the boundary line to show where y < – 3. Holt Algebra 2

2 -5 Linear Inequalities in Two Variables Example 5 Graph 3 x + 4 y ≤ 12 using intercepts. Step 1 Find the intercepts. Substitute x = 0 and y = 0 into 3 x + 4 y = 12 to find the intercepts of the boundary line. y-intercept 3 x + 4 y 3(0) + 4 y 4 y y Holt Algebra 2 = = 12 12 12 3 x-intercept 3 x + 4 y 3 x + 4(0) 3 x x = = 12 12 12 4

2 -5 Linear Inequalities in Two Variables Example 5 Continued Step 2 Draw the boundary line. The line goes through (0, 3) and (4, 0). Draw a solid line for the boundary line because it is part of the graph. Step 3 Find the correct region to shade. Substitute (0, 0) into the inequality. Because 0 + 0 ≤ 12 is true, shade the region that contains (0, 0). Holt Algebra 2 (0, 3) (4, 0)

2 -5 Linear Inequalities in Two Variables Example 6 Graph 3 x – 4 y > 12 using intercepts. Step 1 Find the intercepts. Substitute x = 0 and y = 0 into 3 x – 4 y = 12 to find the intercepts of the boundary line. y-intercept 3 x – 4 y = 12 3(0) – 4 y = 12 y=– 3 Holt Algebra 2 x-intercept 3 x – 4 y = 12 3 x – 4(0) = 12 3 x = 12 x=4

2 -5 Linear Inequalities in Two Variables Example 7: Problem-Solving Application A school carnival charges $4. 50 for adults and $3. 00 for children. The school needs to make at least $135 to cover expenses. A. Using x as the adult ticket price and y as the child ticket price, write and graph an inequality for the amount the school makes on ticket sales. B. If 25 child tickets are sold, how many adult tickets must be sold to cover expenses? Holt Algebra 2

2 -5 2 Linear Inequalities in Two Variables Make a Plan Let x represent the number of adult tickets and y represent the number of child tickets that must be sold. Write an inequality to represent the situation. Adult price 4. 50 times number of adult tickets • x plus child price times number of child tickets + 3. 00 • y is at least total. 135 An inequality that models the problem is 4. 5 x + 3 y ≥ 135. Holt Algebra 2

2 -5 3 Linear Inequalities in Two Variables Solve Find the intercepts of the boundary line. 4. 5(0) + 3 y = 135 4. 5 x + 3(0) = 135 y = 45 Graph the boundary line through (0, 45) and (30, 0) as a solid line. Shade the region above the line that is in the first quadrant, as ticket sales cannot be negative. Holt Algebra 2 x = 30

2 -5 Linear Inequalities in Two Variables If 25 child tickets are sold, 4. 5 x + 3(25) ≥ 135 4. 5 x + 75 ≥ 135 _ 4. 5 x ≥ 60, so x ≥ 13. 3 Substitute 25 for y in 4. 5 x + 3 y ≥ 135. Multiply 3 by 25. A whole number of tickets must be sold. At least 14 adult tickets must be sold. 14($4. 50) + 25($3. 00) = $138. 00, so the answer is reasonable. Holt Algebra 2

2 -5 Linear Inequalities in Two Variables Example 8 A café gives away prizes. A large prize costs the café $125, and the small prize costs $40. The café will not spend more than $1500. How many of each prize can be awarded? How many small prizes can be awarded if 4 large prizes are given away? Holt Algebra 2

2 -5 3 Linear Inequalities in Two Variables Solve Find the intercepts of the boundary line. 40(0) + 125 y = 1500 y = 12 40 x + 125(0) = 1500 x = 37. 5 Graph the boundary line through (0, 12) and (37. 5, 0) as a solid line. Shade the region below the line that is in the first quadrant, as prizes awarded cannot be negative. Holt Algebra 2

2 -5 Linear Inequalities in Two Variables Bellwork 2. 5 1. Graph 2 x – 5 y 10 using intercepts. 2. Solve – 6 y < 18 x – 12 for y. Graph the solution. y > – 3 x + 2 Holt Algebra 2

2 -5 Linear Inequalities in Two Variables Bellwork 2. 5 3. Potatoes cost a chef $18 a box, and carrots cost $12 a box. The chef wants to spend no more than $144. Use x as the number of boxes of potatoes and y as the number of boxes of carrots. a. Write an inequality for the number of boxes the chef can buy. 18 x + 12 y ≤ 144 b. How many boxes of potatoes can the chef order if she orders 4 boxes of carrot? no more than 5 Holt Algebra 2