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

2. 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.

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

4. 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.

5. 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 > (0) – 4 ?
0 > – 4

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

7. 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).

8. 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.

9. 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 ≤ –x The graph is the red region inside and including the parabola y = –x2 + 4.

10. 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.

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

12. 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.