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# 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: $x^2 < 9$

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

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

Find the critical values

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

Work out the positive inequality

If $x^2<9$then $x$ must be **less than** $3$. This is because $3^2=9$, so anything less than 9 must have a smaller square root.

Work out the negative inequality

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

Put these together

$-3 < x < 3$

Solve$x^2 < 4$

Find the solution to $x^2 > 16$

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

$0 > x^2 + 3x + 2$

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

Solve $0 > x^2 + 3x + 2$

Make it an equation equal to 0

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

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

Solve $(x+1)=0$

Solve $(x+2)=0$

These are the critical values

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

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

The critical values are 0 and -2. Solve:

$x^2 + 2x < 0$