## Solving Absolute Inequalities

It is very easy to solve linear inequalities of the form almost as easy as solving linear equations: Solving an absolute inequality, where the equation includes modulus signs is a little bit trickier. When we remove the minus sign it can be hard to work out which way the inequality signs point. Given this, the safest way to solve these equations is either to sketch both absolute functions so that you can see graphically which way the modulus signs point, or square both sides, making both sides positive even without the need for the modulus brackets, then factorising and solving the resulting quadratic inequality.

Example: Solve Squaring both sides gives We expand the brackets and simplify to give Now move every term to the right hand side to give We can divide by the common factor non zero 3 to obtain The quadratic factorises to give The graph is illustrated below. We want those values of for which ie hence the set of values of satisfying is given by If instead the question had asked to solve we would have had solved so that or Notice that now the set of consists of two intervals. 