Quadratic Inequalities. How to solve a quadratic inequality by drawing out the quadratic graph.

In order to solve a quadratic inequality, first turn the quadratic inequality into a quadratic equation. Once this is done, you can solve this quadratic equation either by factorising or by the quadratic formula.

All you have to do now is get the inequality signs the right way round in your answer. You can do this by sketching out the graph of the quadratic equation and seeing which values lie above the x axis or which values lie below the x axis. If the inequality sign is > or ≥ then you are looking for the y values that lie above the x-axis and if the inequality sign is < or ≤ then you are looking for y values that lie under the x axis.

Let’s take a look at some examples of solving quadratic inequalities.

Example 1

Solve the quadratic inequality x² + 3x – 10 < 0.

First make this a quadratic equation.

x² + 3x – 10 = 0

Next factorise this quadratic and solve:

(x-2)(x+5) = 0

So either x-2 = 0 or x+5 = 0.

So the two solutions are x = 2 or x = -5.

So next draw a graph of this quadratic equation.

Since the inequality sign is less than, then the values you need are under the x axis. These are the values between x = -5 and x= 2.

So x > -5 and x < 2. You can also write this as -5 < x < 2.

Example 2

Solve the quadratic inequality x² + 3x > 0

Like example 1,first make this a quadratic equation.

x² + 3x = 0

Next factorise this quadratic and solve:

x(x+3) = 0

So either x = 0 or x+ 3 = 0.

So the two solutions are x = 0 or x = -3.

So next draw a graph of this quadratic equation.

Since the inequality sign is more than, then the values you need are above the x axis. These are the values to the left of -3 or to the right of 0.

So x < -3 or x > 0.

Let’s have a look at one final example.

Example 3

Solve 2x - x² ≥ 0

First make this into a quadratic equation and solve:

x(2-x) = 0

So the two solutions of this quadratic equation are either x = 0 or x = 2.

So next draw a graph of this quadratic equation.

Since the inequality sign is more than or equal to, then the values you need are on or above the x-axis. These are the values between x =0 and x = 2.

So the solution set is x ≥ 0 and x ≤ 2, which can be written as 0 ≤ x ≤ 2.

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