How Solve a Quadratic Inequality

Quick review of the graphs and equations of linear inequalities.

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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 graph on the left expresses the equation of the parabola y= x² −1. From the picture you can easily determine what the solutions are.

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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 of a quadratic inequality are all of the points within the area y > X² −1 where y = 0. In other words, the solution of a quadratic equation holds the same meaning that you are accustomed to. The solution is just where the graph crosses the X-axis. The new twist is that instead of just two or fewer points. A quadratic inequality represents all the points where the shaded in region crosses the axis.

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

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What about the solutions to the quadratic inequality

y < X² − 1



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In this case we are looking for all points on the x-axis that are included in the graph of y < X² − 1. Graphically, all of these points are the points where y = 0 that extend outwards from the solutions of the parabola y= X² − 1

See Graphs side by side (Easier to compare graphs)

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 r₁ and r₂.

The Greater Than Inequality	The Less Than Inequality
0 > ax² + bx + c	0 < ax² + bx + c
Solution: {x| r1 < X < r2 }	Solution: {x| x < r1 or x > r2 }

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

The Greater Than Inequality	The Less Than Inequality
0 ≥ ax² + bx + c	0 ≤ ax² + bx + c
Solution: {x| r1 ≤ X ≤ r2 }	Solution: {x| x ≤ r1 or x ≥ r2 }

Warning about imaginary solutions:

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