Find the set of values of x for which x(x-4) > 12
When solving this quadratic inequality, we should use a three-step approach. The first step is to reformat the inequality to put one side equal to zero, which is the standard format of a quadratic equation. Once this is done, we can re-imagine the inequality as an equation, by substituting the ">" sign for an "=" sign which can be solved. Finally, these solutions of x must then be used as the critical values which satisfy the inequality - to ensure the correct inequality is given, we can imagine a graphical plot of the quadratic equation, and find the regions which satisfy our inequality. In this example, our inequality x(x-4) > 12 can be rewritten as x2 - 4x - 12 > 0 by expanding the brackets and rearranging. By imagining this inequality as an equation, we obtain x2 - 4x - 12 = 0, which we can solve by factorising to obtain the critical values of our final solution. This is solved by rewriting the equation as (x-6)(x+2) = 0, giving the values of -2 and 6 that satisfy the equation. Finally, we must use these values as the critical values for our solution. If we were to draw the function f(x) = x2 - 4x - 12 graphically, we would see that the regions for which the graph is above the value of 0, and thus satisfy our inequality x2 - 4x - 12 > 0, are below -2 and above 6, rather than between these critical values. Hence, our final solution is x < -2, x > 6.
