Objective I will graph quadratic inequalities similarly to quadratic equations in order to solve quadratic inequalities.

Quadratics Before we get started let’s review. A quadratic equation is an equation that can be written in the form , where a, b and c are real numbers and a cannot equal zero. In this lesson we are going to discuss quadratic inequalities.

Quadratic Inequalities What do they look like? Here are some examples:

Quadratic Inequalities When solving inequalities we are trying to find all possible values of the variable which will make the inequality true. Consider the inequality We are trying to find all the values of x for which the quadratic is greater than zero or positive.

Graph a quadratic inequality EXAMPLE 1 Graph a quadratic inequality Graph y > x2 + 3x – 4. SOLUTION STEP 1 Graph y = x2 + 3x – 4. Because the inequality symbol is >, make the parabola dashed. STEP 2 Test a point inside the parabola, such as (0, 0). y > x2 + 3x – 4 0 > 02 + 3(0) – 4 ? 0 > – 4

EXAMPLE 1 Graph a quadratic inequality So, (0, 0) is a solution of the inequality. STEP 3 Shade the region inside the parabola.

EXAMPLE 2 Use a quadratic inequality in real life Rappelling A manila rope used for rappelling down a cliff can safely support a weight W (in pounds) provided W ≤ 1480d2 where d is the rope’s diameter (in inches). Graph the inequality. SOLUTION Graph W = 1480d2 for nonnegative values of d. Because the inequality symbol is ≤, make the parabola solid. Test a point inside the parabola, such as (1, 2000).

Use a quadratic inequality in real life EXAMPLE 2 Use a quadratic inequality in real life W ≤ 1480d2 2000 ≤ 1480(1)2 ? 2000 ≤ 1480 Because (1, 2000) is not a solution, shade the region below the parabola.

Graph a system of quadratic inequalities EXAMPLE 3 Graph a system of quadratic inequalities Graph the system of quadratic inequalities. y < –x2 + 4 Inequality 1 y > x2 – 2x – 3 Inequality 2 SOLUTION STEP 1 Graph y ≤ –x2 + 4. The graph is the red region inside and including the parabola y = –x2 + 4.

EXAMPLE 3 Graph a system of quadratic inequalities STEP 2 Graph y > x2 – 2x – 3. The graph is the blue region inside (but not including) the parabola y = x2 – 2x – 3. STEP 3 Identify the purple region where the two graphs overlap. This region is the graph of the system.

GUIDED PRACTICE for Examples 1, 2, and 3 Graph the inequality. y > x2 + 2x – 8 y < 2x2 – 3x + 1

Independent Practice for Examples 1, 2, and 3 Graph the inequality. 1. y < –x2 + 4x + 2 2. Graph the system of inequalities consisting of y ≥ x2 and y < –x2 + 5.