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.

02 cbxax

Quadratic Inequalities

What do they look like?

Here are some examples:

0732 xx

0443 2 xx

162 x

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.

062 xx

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.

Test a point inside the parabola, such as (0, 0).

STEP 2

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

A manila rope used for rappelling down a cliff can safely support a weight W (in pounds) provided

Rappelling

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).

EXAMPLE 2 Use a quadratic inequality in real life

W ≤ 1480d2

2000 ≤ 1480

Because (1, 2000) is not a solution, shade the region below the parabola.

2000 ≤ 1480(1)2?

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.

Identify the purple region where the two graphs overlap. This region is the graph of the system.

STEP 3

GUIDED PRACTICE for Examples 1, 2, and 3

Graph the inequality.

1. y > x2 + 2x – 8 y < 2x2 – 3x + 12.

Independent Practice for Examples 1, 2, and 3

Graph the inequality.

y < –x2 + 4x + 21. 2. Graph the system of inequalities consisting of y ≥ x2 and y < –x2 + 5.