Subtracting

Subtracting works like adding, with one trap. Take $\frac{4}{x-2}-\frac{1}{x+3}$. Over the common denominator $\left(x-2\right)\left(x+3\right)$, the tops are $4\left(x+3\right)$ and $x-2$, and you take away the whole second top:

$$4x+12-\left(x-2\right) = 4x+12-x+2$$

So the difference is $\frac{3x+14}{\left(x-2\right)\left(x+3\right)}$. The fraction bar groups its top the way brackets do, so the minus reaches every term of it, and the minus 2 turns into plus 2. Forget that and you get $3x+10$. Try x = 3 to see which is right: $\frac{4}{1}-\frac{1}{6}$ is 23/6, the correct top gives 23 over 6, and the wrong one gives 19 over 6.