# ΙΒ μαθηματικά – Inequalities

##### Inequalities – IB mathematics HL

Find the set of values of x for which

$$\frac{x}{2x-5}>4$$

Solution

The rules for solving equations also apply to inequalities with the main difference that when we multiply or divide both sides of an inequality by a negative number we must reverse its sign.

$$\frac{x}{2x-5}>4\Leftrightarrow\frac{x}{2x-5}-4>0$$ $$\frac{x-4(2x-5)}{2x-5}>0 \Leftrightarrow\frac{x-8x+20}{2x-5}>0$$ $$\frac{-7x+20}{2x-5}>0 \Leftrightarrow(-7x+20)(2x-5)>0$$

Now, we have a quadratic inequality which has solutions

$$-\frac{5}{2}<x<\frac{20}{7}$$