4.9: Graph and Solve Quadratic Inequalities Objectives: Solve quadratic inequalities by using tables and graphs. Solve quadratic inequalities by using algebra. Common Core Standards: A-CED-1, F-IF-2

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).

In Lesson 2-5, you solved linear inequalities in two variables by graphing. You can use a similar procedure to graph quadratic inequalities. y ax 2 + bx + c y ≤ ax 2 + bx + c y ≥ ax 2 + bx + c

Graph y ≥ x 2 – 7x + 10.

Graph: y ≥ 2x 2 – 5x – 2

y < –3x 2 – 6x – 7

Graphing Systems of Linear Inequalities. Graph and Solve y < -x y > x 2 – 2x - 3

Quadratic inequalities in one variable, such as ax 2 + 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

Solve the inequality by using tables or graphs. Example 2A: Solving Quadratic Inequalities by Using Tables and Graphs x 2 + 8x + 20 ≥ 5 Use a graphing calculator to graph each side of the inequality. Set Y 1 equal to x 2 + 8x + 20 and Y 2 equal to 5. Identify the values of x for which Y 1 ≥ Y 2.

Example 2A Continued The parabola is at or above the line when x is less than or equal to –5 or greater than or equal to –3. So, the solution set is x ≤ –5 or x ≥ –3 or (–∞, –5] U [–3, ∞). The table supports your answer. –6 –4 – The number line shows the solution set.

Solve the inequality by using tables or graph. x 2 + 8x + 20 < 5

Solve Algebraicly x 2 + 2x > 15