Solving Absolute Inequalities

It is very easy to solve linear inequalities of the formalmost 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 givesWe expand the brackets and simplify to giveNow move every term to the right hand side to giveWe can divide by the common factor non zero 3 to obtainThe quadratic factorises to giveThe graphis illustrated below.

We want those values offor whichiehence the set of values ofsatisfyingis given by

If instead the question had asked to solvewe would have had solved so thatorNotice that now the set ofconsists of two intervals.

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