Before learning how to solve the equation of a quadratic inequality. It is important that you:

- can graph parabolas
- understand the vertex and standard form equation of a parabola
- solve quadratic equations

Quick review of the graphs and equations of linear inequalities.

The graphs of quadratic inequalities follow the same general relationship. Greater than inequalities are the region above the equation's graph and less than inequalities are made up by the region underneath the graph of the equation.

#### What is the solution of a quadratic inequality?

In the end, you can use a pretty straight forward formula to solve quadratic inequalities. First, it's important to try to understand what a quadratic inequality is and also to remember what the solution of a quadratic equation is.

**Steps to solve**

a quadratic inequality

The solutions are the two points where the quadratic equation crosses the x-axis. The same concept of quadratic solution applies to quadratic inequalities.

Let's examine the graph of the quadratic inequality, y > X² − 1

The solution is all points on the x-axis between 1 and -1.

What about the solutions to the quadratic inequality y < X² − 1

A closer look at Y > X² − 1

When we are trying to calculate the solution of a quadratic inequality Y > X² − 1. We want to find all values of x for which y =0 & Y > X² − 1.

To calculate the solution, we must determine when:

- 0 > X² − 1
- 0 > (x − 1)(x + 1)
- Solution set {X| −1 < x < 1 }

A closer look at Y < X² − 1

When we are trying to calculate the solution of a quadratic inequality Y < X² − 1. We want to find all values of x for which y =0 & Y < X² − 1.

To calculate the solution, we must determine when:

- 0 < X² − 1
- 0 < (x − 1)(x + 1)
- Solution set {X| X < −1 or X > 1}

**General Formula**

for Solutions of Quadratic Inequalities

The table below represents two general formulas that express the solution of a quadratic inequality of a parabola that opens upwards (ie a > 0) whose roots are r_{1} and r_{2}.

**The Greater Than Inequality**

0 > ax² + bx + c

**Solution:** {x| r_{1} < X < r_{2} }

**The Less Than Inequality**

0 < ax² + bx + c

**Solution:** {x| x < r_{1} or x > r_{2} }

We can reproduce these general formula for inequalities that include the quadratic itself (ie ≥ and ≤)

**The Greater Than Inequality**

0 ≥ ax² + bx + c

**Solution:** {x| r_{1} ≤ X ≤ r_{2} }

**The Less Than Inequality**

0 ≤ ax² + bx + c

**Solution:** {x| x ≤ r_{1} or x ≥ r_{2} }

Although the solution of a quadratic equation could be imaginary The solution of a quadratic inequality cannot include imaginary numbers--this is becuase imaginary numbers cannot be ordered

**Practice:**

How to graph and solve a quadratic inequality