# Inequalities

### Examples

1. Solve the inequality $2x+3\textgreater 5x-6$. Linear inequalities can be solved in very much the same way as linear equations – effectively making x the subject. Take 2x from both sides of the inequality: $3\textgreater 3x-6$ then add 6 to both sides of the inequality $9\textgreater 3x$. Dividing both sides by 3 gives the solution x>3.
2. Solve the quadratic inequality $x^2-x\textgreater 6$. Quadratic inequalities are solved in very much the same way as quadratic equations – setting one side to 0. It is very important to remember here that quadratic equations have equality – you are looking for roots i.e. where the quadratic cuts the x-axis. When solving quadratic inequalities you are not looking for roots, you are looking to see where the graph is positive or negative. Finding the roots is helpful as it tells you where the graph changes from being positive to negative or vice versa. $x^2-x-6\textgreater 0$ can be factorised to $(x-3)(x+2)\textgreater 0$. The roots would be x=3 and x=-2 meaning that the graph is positive if we choose any x less than -2 or x more than 3. The solution is x>3 or x<-2.