HSC Extension 1 and 2 Mathematics/3-Unit/Preliminary/Other inequalities
In Mathematics Extension 1, higher-order inequalities are introduced. I have written down the steps that need to be taken in order to solve these challenging problems.
When you have a rational function as an inequality (e.g. f(x)/g(x)>a), g(x) might be negative, so you cannot simply multiply g(x) by both sides and cancel out.
There are serveral methods for this kind of problems.
The first method to use is to subtract 'a' to both sides so that you get (f(x)/g(x))-a>0
Rearrange the expression so that there is a common denominator and a single rational function.
After you rearrange the expression, multiply the denominator by the numerator. This would be your 'sign graph'. Using your polynomial skills learnt from Year 10 Advanced Mathematics, perform a rough sketch of the function. Be sure to write down the 'regions' in which the polynomial is greater than 0 (or less than 0, depending on the question). These would be the solutions to your inequality.
You would also have to be careful with the original rational function's denominator and see when the function is undefined. For example if the denominator is x+1, -1 should not be in the solution set of the inequality, because the function will be undefined (i.e. it would be a vertical asymptote if you sketch the function).
The second method is to multiply both sides with the square of the denominator [g(x)]2. By this way, you do not need to worry whether g(x) is negative or not since the square of anything is positive. I found this way more effective.
Hope this helps!