5.8 – Solving Quadratic Inequalities Objectives: Write, solve, and graph a quadratic inequality in one variable. Write, solve, and graph a quadratic inequality in two variables. Standard: H. Select and use an appropriate strategy to solve inequalities.

I. One-Variable Quadratic Inequalities You can determine the solution to a given inequality by finding the roots of the related quadratic equation or by using the graph of the related quadratic equation.  Less ThAND GreatOR

1b. x 2 – 8x + 12 ≤ 0 x ≥ smaller root and x ≤ larger root x 2 – 8x + 12 = 0 (x – 6)(x – 2) = 0 x = 6 and x = 2 Therefore, x ≥ 2 and x ≤ 6

Katie makes and sells T-shirts. A consultant found that her monthly costs, C, are related to the selling price, p, of the shirts by the function C(p) = 75p The revenue, R, from the sale of the shirts is represented by R(p) = -25p p. Her profit, P, is the difference between the revenue and the costs each month. P(p) = R(p) – C(p) = -25p p – (75p ) = -25p p – 2500 At what price range can Katie sell her T-shirts in order to make a profit? -25p p – 2500 > 0Divide both sides by -25 p 2 – 25p < 0Factor (p – 20)(p – 5) < 0 p 5

Ex 3. Solve each inequality

II. Two-Variable Quadratic Inequalities  A quadratic inequality in two variables is an inequality that can be written in one of the forms below, where a, b, and c are real numbers and a ≠ 0.  y ≥ ax 2 + bx + c y > ax 2 + bx + c  y ≤ ax 2 + bx + c y < ax 2 + bx + c

b. y < (x – 1) 2 – 5 c. y ≤ (x + 2) 2 - 3