Inequalities

When dealing with inequalities, it is important to remember that if you divide or multiply by a negative number, the direction of the inequality is changed.

When solving quadratic inequalities (inequalities with x² in them), it is necessary to analyse the various cases to solve the inequality.

Example

Solve x² + 3x + 2 > 0

Now, we can factorise the left hand side to get:

(x + 2)(x + 1) > 0

To solve this, we need to divide it up into a number of cases, based upon when the sign of one of the factors changes:

So when x < -2, for example, we know that (x + 2) is less than 0 and (x + 1) is less than zero. Therefore (x + 2)(x + 1) > 0 (since negative times negative = positive).

From the table, therefore, (x + 2)(x + 1) > 0 when x < -2 or x > -1 .

Drawing a table like this makes things easier.

Alternatively, you can draw the graph of x² + 3x + 2 > 0 and look at where the graph lies above the x-axis. You should get the same answer.

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