Cross-Multiplying

When each side is a single fraction, there is a shortcut. In $\frac{4}{x+2} = \frac{3}{x-1}$, multiply each top by the other side's bottom and set the products equal. This is cross multiplying:

$$4\left(x-1\right) = 3\left(x+2\right)$$

Why does it work? It is the same as multiplying both sides by both bottoms, $\left(x+2\right)\left(x-1\right)$. On each side one bottom cancels and the other stays, landing next to the opposite top. It only works when each side is one fraction: $\frac{1}{x}+2 = \frac{3}{x}$ has two terms on the left, so you clear it the long way.

From there it is linear: $4x-4 = 3x+6$, so x is 10. The bottoms held x, so check it: at 10 neither $x+2$ nor $x-1$ is 0, and both sides are 1/3.