Solve the inequality x^2 + 5x -24 ≥ 0.
There are a few different ways to approach this problem but the simplest and most easy to visualise solution comes from sketching the curve y = x2 + 5x - 24 and identifying the range of x-values for which y ≥ 0. The first step is to factorise the equation to find the points where the curve crosses the x-axis and moves from being greater than 0 to less than 0 or vice versa. In this example you need to find two numbers that add to make 5 and multiply to make -24, namely -3 and 8. The equation can then be written as y = (x + 8)(x - 3) and the points where the curve crosses the x axis are x = -8 and x = 3. Since the x2 term is positive, we know that this graph has a minimum rather than a maximum and so the regions where y ≥ 0 lie before it crosses the x-axis for the first time and after it crosses it for the second time. The solution is therefore: x ≥ 3 and x ≤ -8.
