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Warm Up 1. Graph the inequality y < 2x + 1. Solve using any method. 2. x 2 – 16x + 63 = 0 3. 3x 2 + 8x = 3 7, 9

Warm Up 1. Graph the inequality y < 2 x + 1

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Warm Up 1. Graph the inequality y < 2 x + 1. Solve using any method. 2. x 2 – 16 x + 63 = 0. 7, 9. 3. 3 x 2 + 8 x = 3. Objectives. Solve quadratic inequalities by using tables and graphs. Solve quadratic inequalities by using algebra. Vocabulary. - PowerPoint PPT Presentation

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Page 1: Warm Up 1.  Graph the inequality y  < 2 x  + 1

Warm Up1. Graph the inequality y < 2x + 1.

Solve using any method.2. x2 – 16x + 63 = 0

3. 3x2 + 8x = 3

7, 9

Page 2: Warm Up 1.  Graph the inequality y  < 2 x  + 1

Solve quadratic inequalities by using tables and graphs.

Solve quadratic inequalities by using algebra.

Objectives

Page 3: Warm Up 1.  Graph the inequality y  < 2 x  + 1

quadratic inequality in two variables

Vocabulary

Page 4: Warm Up 1.  Graph the inequality y  < 2 x  + 1

Many business profits can be modeled by quadratic functions. To ensure that the profit is above a certain level, financial planners may need to graph and solve quadratic inequalities.

A quadratic inequality in two variables can be written in one of the following forms, where a, b, and c are real numbers and a ≠ 0. Its solution set is a set of ordered pairs (x, y).

Page 5: Warm Up 1.  Graph the inequality y  < 2 x  + 1

In Lesson 2-5, you solved linear inequalities in two variables by graphing. You can use a similar procedure to graph quadratic inequalities.

y < ax2 + bx + c y > ax2 + bx + c

y ≤ ax2 + bx + c y ≥ ax2 + bx + c

Page 6: Warm Up 1.  Graph the inequality y  < 2 x  + 1

Graph y ≥ x2 – 7x + 10.

Example 1: Graphing Quadratic Inequalities in Two Variables

Step 1 Graph the boundary of the related parabola y = x2 – 7x + 10 with a solid curve. Its y-intercept is 10, its vertex is (3.5, –2.25), and its x-intercepts are 2 and 5.

Page 7: Warm Up 1.  Graph the inequality y  < 2 x  + 1

Example 1 Continued

Step 2 Shade above the parabola because the solution consists of y-values greater than those on the parabola for corresponding x-values.

Page 8: Warm Up 1.  Graph the inequality y  < 2 x  + 1

Example 1 Continued

Check Use a test point to verify the solution region.

y ≥ x2 – 7x + 10

0 ≥ (4)2 –7(4) + 10

0 ≥ 16 – 28 + 10

0 ≥ –2

Try (4, 0).

Page 9: Warm Up 1.  Graph the inequality y  < 2 x  + 1

Graph the inequality.

Step 1 Graph the boundary of the related parabola

y = 2x2 – 5x – 2 with a solid curve. Its y-intercept is –2, its vertex is (1.3, –5.1), and its x-intercepts are –0.4 and 2.9.

Check It Out! Example 1a

y ≥ 2x2 – 5x – 2

Page 10: Warm Up 1.  Graph the inequality y  < 2 x  + 1

Step 2 Shade above the parabola because the solution consists of y-values greater than those on the parabola for corresponding x-values.

Check It Out! Example 1a Continued

Page 11: Warm Up 1.  Graph the inequality y  < 2 x  + 1

Check Use a test point to verify the solution region.

y < 2x2 – 5x – 2

0 ≥ 2(2)2 – 5(2) – 2

0 ≥ 8 – 10 – 2

0 ≥ –4

Try (2, 0).

Check It Out! Example 1a Continued

Page 12: Warm Up 1.  Graph the inequality y  < 2 x  + 1

Graph each inequality.

Step 1 Graph the boundary of the related parabola y = –3x2 – 6x – 7 with a dashed curve. Its y-intercept is –7.

Check It Out! Example 1b

y < –3x2 – 6x – 7

Page 13: Warm Up 1.  Graph the inequality y  < 2 x  + 1

Step 2 Shade below the parabola because the solution consists of y-values less than those on the parabola for corresponding x-values.

Check It Out! Example 1b Continued

Page 14: Warm Up 1.  Graph the inequality y  < 2 x  + 1

Check Use a test point to verify the solution region.

y < –3x2 – 6x –7

–10 < –3(–2)2 – 6(–2) – 7

–10 < –12 + 12 – 7

–10 < –7

Try (–2, –10).

Check It Out! Example 1b Continued

Page 15: Warm Up 1.  Graph the inequality y  < 2 x  + 1

Quadratic inequalities in one variable, such as ax2 + bx + c > 0 (a ≠ 0), have solutions in one variable that are graphed on a number line.

For and statements, both of the conditions must be true. For or statements, at least one of the conditions must be true.

Reading Math

Page 16: Warm Up 1.  Graph the inequality y  < 2 x  + 1

Solve the inequality by using tables or graphs.

Example 2A: Solving Quadratic Inequalities by Using Tables and Graphs

x2 + 8x + 20 ≥ 5

Use a graphing calculator to graph each side of the inequality. Set Y1 equal to x2 + 8x + 20 and Y2 equal to 5. Identify the values of x for which Y1 ≥ Y2.

Page 17: Warm Up 1.  Graph the inequality y  < 2 x  + 1

Solve the inequality by using tables and graph.

Example 2B: Solving Quadratics Inequalities by Using Tables and Graphs

x2 + 8x + 20 < 5

Use a graphing calculator to graph each side of the inequality. Set Y1 equal to x2 + 8x + 20 and Y2 equal to 5. Identify the values of which Y1 < Y2.

Page 18: Warm Up 1.  Graph the inequality y  < 2 x  + 1

Solve the inequality by using tables and graph.

x2 – x + 5 < 7

Use a graphing calculator to graph each side of the inequality. Set Y1 equal to x2 – x + 5 and Y2 equal to 7. Identify the values of which Y1 < Y2.

Check It Out! Example 2a

Page 19: Warm Up 1.  Graph the inequality y  < 2 x  + 1

Solve the inequality by using tables and graph.

2x2 – 5x + 1 ≥ 1 Use a graphing calculator to graph each side of the inequality. Set Y1 equal to 2x2 – 5x + 1 and Y2 equal to 1. Identify the values of which Y1 ≥ Y2.

Check It Out! Example 2b

Page 20: Warm Up 1.  Graph the inequality y  < 2 x  + 1

Solve the inequality by using algebra.

Step 1 Write the related equation.

Check It Out! Example 3a

x2 – 6x + 10 ≥ 2

x2 – 6x + 10 = 2

Page 21: Warm Up 1.  Graph the inequality y  < 2 x  + 1

Write in standard form.

Step 2 Solve the equation for x to find the critical values.

x2 – 6x + 8 = 0

x – 2 = 0 or x – 4 = 0

(x – 2)(x – 4) = 0 Factor.

Zero Product Property.

Solve for x.x = 2 or x = 4

The critical values are 2 and 4. The critical values divide the number line into three intervals: x ≤ 2, 2 ≤ x ≤ 4, x ≥ 4.

Check It Out! Example 3a Continued

Page 22: Warm Up 1.  Graph the inequality y  < 2 x  + 1

Step 3 Test an x-value in each interval.

(1)2 – 6(1) + 10 ≥ 2

x2 – 6x + 10 ≥ 2

(3)2 – 6(3) + 10 ≥ 2

(5)2 – 6(5) + 10 ≥ 2

Try x = 1.

Try x = 3.

Try x = 5.

Check It Out! Example 3a Continued

x

–3 –2 –1 0 1 2 3 4 5 6 7 8 9

Critical values

Test points

Page 23: Warm Up 1.  Graph the inequality y  < 2 x  + 1

Shade the solution regions on the number line. Use solid circles for the critical values because the inequality contains them. The solution is x ≤ 2 or x ≥ 4.

–3 –2 –1 0 1 2 3 4 5 6 7 8 9

Check It Out! Example 3a Continued

Page 24: Warm Up 1.  Graph the inequality y  < 2 x  + 1

Solve the inequality by using algebra.

Step 1 Write the related equation.

Check It Out! Example 3b

–2x2 + 3x + 7 < 2

–2x2 + 3x + 7 = 2

Page 25: Warm Up 1.  Graph the inequality y  < 2 x  + 1

Write in standard form.

Step 2 Solve the equation for x to find the critical values.

–2x2 + 3x + 5 = 0

–2x + 5 = 0 or x + 1 = 0

(–2x + 5)(x + 1) = 0 Factor.

Zero Product Property.

Solve for x.x = 2.5 or x = –1

The critical values are 2.5 and –1. The critical values divide the number line into three intervals: x < –1, –1 < x < 2.5, x > 2.5.

Check It Out! Example 3b Continued

Page 26: Warm Up 1.  Graph the inequality y  < 2 x  + 1

Step 3 Test an x-value in each interval.

–2(–2)2 + 3(–2) + 7 < 2

–2(1)2 + 3(1) + 7 < 2

–2(3)2 + 3(3) + 7 < 2

Try x = –2.

Try x = 1.

Try x = 3.

–3 –2 –1 0 1 2 3 4 5 6 7 8 9

Critical values

Test points

Check It Out! Example 3b Continued

x

–2x2 + 3x + 7 < 2

Page 27: Warm Up 1.  Graph the inequality y  < 2 x  + 1

Shade the solution regions on the number line. Use open circles for the critical values because the inequality does not contain or equal to. The solution is x < –1 or x > 2.5.

–3 –2 –1 0 1 2 3 4 5 6 7 8 9

Check It Out! Example 3

Page 28: Warm Up 1.  Graph the inequality y  < 2 x  + 1

A compound inequality such as 12 ≤ x ≤ 28 can be written as {x|x ≥12 U x ≤ 28}, or x ≥ 12 and x ≤ 28. (see Lesson 2-8).

Remember!

Page 29: Warm Up 1.  Graph the inequality y  < 2 x  + 1

Example 4: Problem-Solving Application

The monthly profit P of a small business that sells bicycle helmets can be modeled by the function P(x) = –8x2 + 600x – 4200, where x is the average selling price of a helmet. What range of selling prices will generate a monthly profit of at least $6000?

Page 30: Warm Up 1.  Graph the inequality y  < 2 x  + 1

11 Understand the Problem

Example 4 Continued

The answer will be the average price of a helmet required for a profit that is greater than or equal to $6000.

List the important information:

• The profit must be at least $6000.

• The function for the business’s profit is P(x) = –8x2 + 600x – 4200.

Page 31: Warm Up 1.  Graph the inequality y  < 2 x  + 1

22 Make a Plan

Write an inequality showing profit greater than or equal to $6000. Then solve the inequality by using algebra.

Example 4 Continued

Page 32: Warm Up 1.  Graph the inequality y  < 2 x  + 1

Solve33

Write the inequality.

–8x2 + 600x – 4200 ≥ 6000

–8x2 + 600x – 4200 = 6000

Find the critical values by solving the related equation.

Write as an equation.

Write in standard form.

Factor out –8 to simplify.

–8x2 + 600x – 10,200 = 0

–8(x2 – 75x + 1275) = 0

Example 4 Continued

Page 33: Warm Up 1.  Graph the inequality y  < 2 x  + 1

Solve33

Use the Quadratic Formula.

Simplify.

x ≈ 26.04 or x ≈ 48.96

Example 4 Continued

Page 34: Warm Up 1.  Graph the inequality y  < 2 x  + 1

Solve33

Test an x-value in each of the three regions formed by the critical x-values.

10 20 30 40 50 60 70

Critical values

Test points

Example 4 Continued

Page 35: Warm Up 1.  Graph the inequality y  < 2 x  + 1

Solve33

–8(25)2 + 600(25) – 4200 ≥ 6000

–8(45)2 + 600(45) – 4200 ≥ 6000

–8(50)2 + 600(50) – 4200 ≥ 6000

5800 ≥ 6000Try x = 25.

Try x = 45.

Try x = 50.

6600 ≥ 6000

5800 ≥ 6000

Write the solution as an inequality. The solution is approximately 26.04 ≤ x ≤ 48.96.

x

x

Example 4 Continued

Page 36: Warm Up 1.  Graph the inequality y  < 2 x  + 1

Solve33

For a profit of $6000, the average price of a helmet needs to be between $26.04 and $48.96, inclusive.

Example 4 Continued

Page 37: Warm Up 1.  Graph the inequality y  < 2 x  + 1

Look Back44

Enter y = –8x2 + 600x – 4200 into a graphing calculator, and create a table of values. The table shows that integer values of x between 26.04 and 48.96 inclusive result in y-values greater than or equal to 6000.

Example 4 Continued

Page 38: Warm Up 1.  Graph the inequality y  < 2 x  + 1

Homework!

Holt Chapter 5 Section 7Page 370 # 19-27 odd, 34, 35, 38,

43, 442 Additional Problems

Page 39: Warm Up 1.  Graph the inequality y  < 2 x  + 1
Page 40: Warm Up 1.  Graph the inequality y  < 2 x  + 1