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Quadratic Inequalities

Quadratic Inequalities

Quadratic Inequalities

Inequalities can also be used to describe functions where x is greater than 1.

Quadratic inequalities, similar to linear inequalities, give us a range of values as a solution rather than a single solution. However, the difference with quadratic inequalities is that they contain an x^2 term.

Sometimes you will be given an inequality that involves a quadratic expression such as: x2<9x^2 < 9

x2>56x^2>56 is the same as:

Let's have a look at solving our example from above: x2<9x^2 < 9


Find the critical values

In this case, these are the solutions to x2\sqrt{x^2}. Remember that there is a positive and a negative square root, so the critical values are 9=3\sqrt{9}=3 or9=3\sqrt{9}=-3.


Work out the positive inequality

If x2<9x^2<9then xx must be less than 33. This is because 32=93^2=9, so anything less than 9 must have a smaller square root.


Work out the negative inequality

If x2<9x^2<9, then xxmust be greater than 3-3. 32=9-3^2=9. Therefore, anything less than 9 should have a square root closer to 0 (and therefore greater).


Put these together

3<x<3 -3 < x < 3

Solvex2<4 x^2 < 4

Find the solution to x2>16x^2 > 16

We may also come across a more complicated quadratic expression on one side of the inequality, such as:

0>x2+3x+20 > x^2 + 3x + 2

We can look to find the critical values by making it an equation and solving by factorising or using the quadratic formula.

Solve 0>x2+3x+20 > x^2 + 3x + 2


Make it an equation equal to 0

To do this, we need to replace the >> sign with an == sign:x2+3x+2=0x^2 + 3x + 2=0


Factorise x2+3x+2=0x^2 + 3x + 2=0


Solve (x+1)=0(x+1)=0


Solve (x+2)=0(x+2)=0


These are the critical values

x=1x=-1 and x=2x=-2. We can use the critical values to identify the solutions on a graph.

Find critical values for x2+2x<0x^2 + 2x < 0

The critical values are 0 and -2. Solve:

x2+2x<0x^2 + 2x < 0