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**A linear inequality in two variables relates two variables using an inequality symbol.**

Example: y > 2x – 4. Its graph is a region of the coordinate plane bounded by a line. The line is called a boundary line, which divides the coordinate plane into two regions. Remember: ≥ and ≤ use solid lines > and < use dashed lines

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To graph y ≥ 2x – 4, make the boundary line solid, and shade the region above the line. To graph y > 2x – 4, make the boundary line dashed because y-values equal to 2x – 4 are not included.

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**Think of the underlines in the symbols ≤ and ≥ as representing solid lines on the graph.**

Helpful Hint

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**Example 1A: Graphing Linear Inequalities**

Graph the inequality Step one: Draw the line but with a dashed line y-intercept = 2 and a slope of . Then shade the region above the boundary line to show .

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

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**Example 1B: Graphing Linear Inequalities**

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. .

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Example 1B Continued 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.

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If the equation of the boundary line is not in slope-intercept form, you can choose a test point that is not on the line to determine which region to shade. If the point satisfies the inequality, then shade the region containing that point. Otherwise, shade the other region. The point (0, 0) is the easiest point to test if it is not on the boundary line. Helpful Hint

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**Example 2: Graphing Linear Inequalities Using Intercepts**

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-intercept x-intercept 3x + 4y = 12 3x + 4y = 12 3(0) + 4y = 12 3x + 4(0) = 12 4y = 12 3x = 12 y = 3 x = 4

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**Step 2 Draw the boundary line. **

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. (0, 3) 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). (4, 0)

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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

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**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. List the important information: The school sells tickets at $4.50 for adults and $3.00 for children. The school needs to make at least $135.

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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. 4.5x + 3y ≥ 135. Find the intercepts of the boundary line. 4.5(0) + 3y = 135 y = 45 4.5x + 3(0) = 135 x = 30

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**If 25 child tickets are sold,**

B. If 25 child tickets are sold, how many adult tickets must be sold to cover expenses? If 25 child tickets are sold, 4.5x + 3(25) ≥ 135 Substitute 25 for y in 4.5x + 3y ≥ 135. 4.5x + 75 ≥ 135 Multiply 3 by 25. 4.5x ≥ 60, so x ≥ 13.3 _ A whole number of tickets must be sold. At least 14 adult tickets must be sold.

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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?

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**Understand the Problem**

1 Understand the Problem The answer will be in two parts: (1) an inequality graph showing the number of each type of prize awarded not too exceed a certain amount (2) the number of small prizes awarded if 4 large prizes are awarded. List the important information: The café awarded large prizes valued at $125 and $40 for small prizes. The café will not spend over $1500.

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2 Make a Plan 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. 1500 y 125 + x 40 total. is less than number awarded times large prize plus Small prize ≤ An inequality that models the problem is 40x + 125y ≤ 135.

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Solve 3 Find the intercepts of the boundary line. 40(0) + 125y = 1500 40x + 125(0) = 1500 y = 12 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.

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**If 4 large prizes are awarded,**

40x + 125(4) ≤ 1500 Substitute 4 for y in 40x + 125y ≤ 135. 40x ≤ 1500 Multiply 125 by 4. A whole number of small prizes must be awarded. 40x ≥ 1000, so x ≤ 25 No more than 25 small prizes can be awarded. Look Back 4 $40(25) + $125(4) = $1500, so the answer is reasonable.

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You can graph a linear inequality that is solved for y with a graphing calculator. Press and use the left arrow key to move to the left side. Each time you press you will see one of the graph styles shown here. You are already familiar with the line style.

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**Example 4: Solving and Graphing Linear Inequalities**

Solve for y. Graph the solution. Multiply both sides by 8x – 2y > 8 –2y > –8x + 8 Subtract 8x from both sides. Divide by –2, and reverse the inequality symbol. y < 4x – 4

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Example 4 Continued Use the calculator option to shade below the line y < 4x – 4. Note that the graph is shown in the STANDARD SQUARE window. ( :ZStandard followed by :ZSquare).

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**Solve 2(3x – 4y) > 24 for y. Graph the solution.**

Check It Out! Example 4 Solve 2(3x – 4y) > 24 for y. Graph the solution. 3x – 4y > 12 Divide both sides by 2. –4y > –3x + 12 Subtract 3x from both sides. Divide by –4, and reverse the inequality symbol.

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**Check It Out! Example 4 Continued**

Use the calculator option to shade below the line . Note that the graph is shown in the STANDARD SQUARE window. ( :ZStandard followed by :ZSquare).

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Lesson Quiz: Part I 1. Graph 2x –5y 10 using intercepts. 2. Solve –6y < 18x – 12 for y. Graph the solution. y > –3x + 2

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Lesson Quiz: Part II 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. 18x + 12y ≤ 144 b. How many boxes of potatoes can the chef order if she orders 4 boxes of carrot? no more than 5

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