Holt Algebra Linear Inequalities in Two Variables 6-6 Linear Inequalities in Two Variables 1. Warm Up (Slide #2) 1. Warm Up (Slide #2) 3. Lesson Presentation (Slide #6 – 33) 3. Lesson Presentation (Slide #6 – 33) 6. Lesson Quiz (Slide #36,37) 6. Lesson Quiz (Slide #36,37) 4. Text Questions (Slide #34) 4. Text Questions (Slide #34) LESSON PLAN 2. Vocab (Slides #4 - 5) 2. Vocab (Slides #4 - 5)

Holt Algebra Linear Inequalities in Two Variables Warm Up Find the intercepts of each line. 1. 3x + 2y = x – y = x + 10 = 2y Write the function in slope-intercept form. Then graph. 4. 2x + 3y = –3

Holt Algebra Linear Inequalities in Two Variables 1.Graph linear inequalities on the coordinate plane 2.Solve problems using linear inequalities Objectives

Holt Algebra Linear Inequalities in Two Variables linear inequality Vocabulary

Holt Algebra Linear Inequalities in Two Variables A linear inequality relates two variables using an inequality symbol such as y > 2x – 4. regiona wedge-shaped area Its graph is a region (…or a wedge-shaped area) of the coordinate plane bounded by a line.

Holt Algebra Linear Inequalities in Two Variables For example:

Holt Algebra Linear Inequalities in Two Variables Think of the underlines in the symbols ≤ and ≥ as representing solid lines on the graph. Helpful Hint

Holt Algebra Linear Inequalities in Two Variables Example 1A: Graphing Linear Inequalities Graph the inequality. The boundary has a y-int. +2 and slope of. Draw the line dashed since it is not part of the solution. Shade the region above the boundary line to show.

Holt Algebra Linear Inequalities in Two Variables Example 1A Continued Check Choose a point in the solution region, such as (3, 2) and test it in the inequality. The test point satisfies the inequality, so the solution region appears to be correct. ? 2 > 1 ?

Holt Algebra Linear Inequalities in Two Variables 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.. Example 1B: Graphing Linear Inequalities

Holt Algebra Linear Inequalities in Two Variables Check The point (0, –2) is a solution because –2 ≤ –1. Note that any point on or below y = –1 is a solution, regardless of the value of x. Example 1B Continued

Holt Algebra Linear Inequalities in Two Variables The boundary line has a y–intercept of ____ and a slope of _____. Draw a ______ line because it is part of the solution. Then shade the region _______ the boundary line to show y > 3x – 2. Check It Out! Example 1a Graph the inequality y ≥ 3x – solid above

Holt Algebra Linear Inequalities in Two Variables Check Is (–3, 2) a solution to the inequality? The test point satisfies the inequality, so the solution region appears to be correct. y ≥ 3x –2 Check It Out! Example 1a Continued 2 ≥ 3(–3) –2 ? 2 ≥ (–9) –2 ? 2 > –11 ?

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

Holt Algebra Linear Inequalities in Two Variables Check It Out! Example 1b Continued Check Is (0, –4) a solution? Yes!

Holt Algebra Linear Inequalities in Two Variables The point (0, 0) is the easiest point to test if it is not on the boundary line. Helpful Hint

Holt Algebra Linear Inequalities in Two Variables Graph 3x + 4y ≤ 12 using intercepts. Example 2: Graphing Linear Inequalities Using Intercepts Step 1 Find the intercepts. y-intercept (is where x = 0)x-intercept (is where y = 0) 3x + 4y = 12 3(0) + 4y = 12 3x + 4(0) = 12 4y = 12 3x + 4y = 12 y = 3 3x = 12 x = 4

Holt Algebra Linear Inequalities in Two Variables Example 2 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 ≤ 12 is true, shade the region that contains (0, 0). (0, 3) (4, 0) Graphing 3x + 4y ≤ 12 using intercepts:

Holt Algebra Linear Inequalities in Two Variables Graph 3x – 4y > 12 using intercepts. Step 1 Find the intercepts. Substitute x = 0 and y = 0 into 3x – 4y = 12 to find the intercepts of the boundary line. y-interceptx-intercept 3x – 4y = 12 3(0) – 4y = 123x – 4(0) = 12 – 4y = 12 3x – 4y = 12 y = – 3 3x = 12 x = 4 Check It Out! Example 2

Holt Algebra Linear Inequalities in Two Variables Step 2 Draw the boundary line. The line goes through (0, –3) and (4, 0). Draw the boundary line dashed because it is not part of the solution. Step 3 Find the correct region to shade. Substitute (0, 0) into the inequality. Because >12 is false, shade the region that does not contain (0, 0). (4, 0) Check It Out! Example 2 (0, –3) Graph 3x – 4y > 12 using intercepts.

Holt Algebra Linear Inequalities in Two Variables Many applications of inequalities in two variables use only nonnegative values for the variables. Graph only the part of the plane that includes realistic solutions. Don’t forget which variable represents which quantity. Caution

Holt Algebra Linear Inequalities in Two Variables Example 3: 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 number of adult tickets and y as the number of child tickets, 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 Linear Inequalities in Two Variables 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. An inequality that models the problem is 4.5x + 3y ≥ y3.00+x4.50 total. is at least number of child tickets times child price plus number of adult tickets times Adult price

Holt Algebra Linear Inequalities in Two Variables Find the intercepts of the boundary line. 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. 4.5(0) + 3y = x + 3(0) = 135 y = 45 x = x + 3y ≥ 135.

Holt Algebra Linear Inequalities in Two Variables If 25 child tickets are sold, At least 14 adult tickets must be sold. 4.5x + 3(25) ≥ x + 75 ≥ x ≥ 60, so x ≥ 13.3 _

Holt Algebra Linear Inequalities in Two Variables Check It Out! Example 3 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 Linear Inequalities in Two Variables Let x represent the number of small prizes and y represent the number of large prizes, the total not too exceed $1500. Write an inequality to represent the situation. An inequality that models the problem is 40x + 125y ≤ y125+x40 total. is less than number awarded times large prize plus number awarded times Small prize ≤

Holt Algebra Linear Inequalities in Two Variables Find the intercepts of the boundary line. 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. 40(0) + 125y = x + 125(0) = 1500 y = 12x = x + 125y ≤ 1500

Holt Algebra Linear Inequalities in Two Variables If 4 large prizes are awarded, No more than 25 small prizes can be awarded. 40x + 125(4) ≤ x ≤ x ≤ 1000 x ≤ 25 40x + 125y ≤ 1500

Holt Algebra Linear Inequalities in Two Variables Graphing a linear inequality with a graphing calculator. Press the key. Press the left arrow key to move to the left side of Each time you press you will see one of the graph styles shown here. ><><

Holt Algebra Linear Inequalities in Two Variables 1.Key “Y=“ 2.Key “<“ twice 3.Key “ENTER” three times to get to the “less than” graphic 4.Key “>” twice to return to the equation area 5.Key “4X-4” 6.Key “GRAPH” Use the calculator graph the solution to y < 4x - 4 Check It Out! Example 4 Continued

Holt Algebra Linear Inequalities in Two Variables 1.Key “Y=“ 2.Key “<“ twice 3.Key “ENTER” three times 4.Key “>” twice 5.Key “(3÷4)X-3” 6.Key “GRAPH” Use the calculator graph the solution to Check It Out! Example 4 Continued

Holt Algebra Linear Inequalities in Two Variables Graph this inequality on a calculator: 2x + 3y < -5 2x + 3y < -5 -2x 3y < -2x – 5 /3 y < -(2/3)x - 5/3 -2x /3 To use the graphing calculator, you must first rewrite the inequality so you are “solving for ‘y’”)