5-7 Solving Quadratic Inequalities. Holt Algebra 2

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2 Example 1: Graphing Quadratic Inequalities in Two Variables Graph f(x) x 2 7x Step 1 Graph the parabola f(x) = x 2 7x + 10 with a solid curve. x f(x)

3 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.

4 Example 1 Continued Check Use a test point to verify the solution region. y x 2 7x (4) 2 7(4) + 10 Try (4, 0)

5 Check It Out! Example 1b Graph each inequality. y < 3x 2 6x 7 Step 1 Graph the boundary of the related parabola y = 3x 2 6x 7 with a dashed curve. x f(x)

6 Check It Out! Example 1b Continued Step 2 Shade below the parabola because the solution consists of y-values less than those on the parabola for corresponding x-values.

7 Check It Out! Example 1b Continued Check Use a test point to verify the solution region. y < 3x 2 6x 7 10 < 3( 2) 2 6( 2) 7 Try ( 2, 10). 10 < < 7

8 Quadratic inequalities can also be solved algebraically. Use factoring or the quadratic formula to find the zeros and then try a test point to see if it is inside or outside the solution region.

9 Example 3: Solving Quadratic Equations by Using Algebra Solve the inequality x 2 10x by using algebra. Step 1 Write the related equation. x 2 10x + 18 = 3

10 Example 3 Continued Step 2 Solve the equation for x to find the critical values. x 2 10x + 21 = 0 (x 3)(x 7) = 0 Factor. x 3 = 0 or x 7 = 0 x = 3 or x = 7 Write in standard form. Zero Product Property. Solve for x. The critical values are 3 and 7. The critical values divide the number line into three intervals: x 3, 3 x 7, x 7.

11 Example 3 Continued Step 3 Test an x-value in each interval. Critical values x 2 10x (2) 2 10(2) (4) 2 10(4) (8) 2 10(8) x x Test points Try x = 2. Try x = 4. Try x = 8.

12 Example 3 Continued Shade the solution regions on the number line. Use solid circles for the critical values because the inequality contains them. The solution is 3 x 7 or [3, 7]

13 Check It Out! Example 3a Solve the inequality by using algebra. x 2 6x Step 1 Write the related equation. x 2 6x + 8 = 2

14 Check It Out! Example 3a Continued Step 2 Solve the equation for x to find the critical values. x 2 6x + 6 = 0 Write in standard form. x = 6 ± x = x = ( 6)2 4(1)(6) 2(1) 6 ± ± 12 2 = 6±2 3 2 x = and x = x = and x = Substitute into the quadratic formula to find the zeros

15 Check It Out! Example 3a Continued Step 3 Test an x-value in the interval. x 2 6x (2) 2 6(2) Try x = 2.

16 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) = 8x x 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?

17 Example 4 Continued 1 Understand the Problem 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) = 8x x 4200.

18 2 Make a Plan Example 4 Continued Write an inequality showing profit greater than or equal to $6000. Then solve the inequality by using algebra.

19 Example 4 Continued 3 Solve Write the inequality. 8x x Find the critical values by solving the related equation. 8x x 4200 = x x 10,200 = 0 8(x 2 75x ) = 0 Write as an equation. Write in standard form. Factor out 8 to simplify.

20 Example 4 Continued 3 Solve Use the Quadratic Formula. Simplify. x or x 48.96

21 Example 4 Continued 3 Solve Test an x-value in each of the three regions formed by the critical x-values. Critical values Test points

22 Example 4 Continued 3 Solve 8(25) (25) (45) (45) (50) (50) x x Try x = 25. Try x = 45. Try x = 50. Write the solution as an inequality. The solution is approximately x

23 Example 4 Continued 3 Solve For a profit of $6000, the average price of a helmet needs to be between $26.04 and $48.96, inclusive.

24 Example 4 Continued 4 Look Back Enter y = 8x x 4200 into a graphing calculator, and create a table of values. The table shows that integer values of x between and inclusive result in y- values greater than or equal to 6000.