### Solving inequalities using the graphical method

Question Sample Titled 'Solving inequalities using the graphical method'

 Given the graph of ${y}={f{{\left({x}\right)}}}$ , we can solve the inequalities ${f{{\left({x}\right)}}}>{k}$ , ${f{{\left({x}\right)}}}={k}$ and ${f{{\left({x}\right)}}}\le{k}$ , where k is a constant, using the graphical method. The procedure is as follows:

 ${\left({1}\right)}$ Draw the straight line ${y}={k}$ on the given graph of ${y}={f{{\left({x}\right)}}}.$ ${\left({2}\right)}$ Find the portion(s) of the graph which lies above (or below) the straight line ${y}={k}$ according to the given inequality. ${\left({3}\right)}$ Find the range of values of ${x}$ that satisfy the given inequality.

 Example Given the graph of ${y}={x}^{{2}}-{3}{x}$ , solve the inequality ${x}^{{2}}-{3}{x}\ge{4}$ graphically. Solution Draw the straight line ${y}={4}$ on the graph of ${y}={x}^{{2}}-{3}{x}$ .

 The two graphs intersect at ${x}=-{1}$ and ${x}={4}$ . When ${x}\le-{1}$ or ${x}\ge{4}$ , the corresponding (red) portions of the graph of ${y}={x}^{{2}}-{3}{x}$ lie on or above the straight line ${y}={4}$ . ∴   The solutions of ${x}^{{2}}-{3}{x}\ge{4}$ are ${x}\le-{1}$ or ${x}\ge{4}$ .

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