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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<9
x2>56 is the same as:
Let's have a look at solving our example from above: x2<9
Find the critical values
In this case, these are the solutions to x2. Remember that there is a positive and a negative square root, so the critical values are 9=3or9=−3.
Work out the positive inequality
If x2<9then x must be less than 3. This is because 32=9, so anything less than 9 must have a smaller square root.
Work out the negative inequality
If x2<9, then xmust be greater than −3. −32=9. Therefore, anything less than 9 should have a square root closer to 0 (and therefore greater).
Put these together
Find the solution to x2>16
We may also come across a more complicated quadratic expression on one side of the inequality, such as:
We can look to find the critical values by making it an equation and solving by factorising or using the quadratic formula.
Make it an equation equal to 0
To do this, we need to replace the > sign with an = sign:x2+3x+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 x2+2x<0
The critical values are 0 and -2. Solve: