Solve the inequality x^2 - 9 > 0
This is a quadratic inequality, because we have an x2 term, so we answer this question by examining the graph of the associated equation y = x2 - 9, and then find out where this graph is greater than 0.So first, we find where the equation, y = x2 - 9 is equal to zero (i.e. y = 0). This is simply solving a quadratic, which we do by factorising and equating each term in brackets to zero, i.e.:x2 - 9 = 0This is the difference of two squares, so the factorisation should be relatively familiar.(x+3)(x-3) = 0Therefore x= - 3 or x = 3.This tells us that the graph crosses the x-axis at x=3 and x=-3.Then, we need to consider what the rest of the graph looks like. Since the equation y = x2 - 9 has a positive x2 term, this graph must be a positive parabola (U-shape) graph. Putting this all together we can sketch a graph of the equation, and now we look back to our original inequality, which asks for when this graph (when the y-values) are greater than zero. We can see this happens when either x > 3 or when x< -3 and so this is our solution.
