How can $\frac{1}{a/x-b/x}$ be equal to $\frac{1}{a-b}$?

Question

In an exercise asking to mark true or false, it shows:

$$\frac{1}{a/x-b/x}=\frac{1}{a-b}$$

It really look like false to me. But the answer is true! How can it be?

Answer 1

Suppose $x\neq 0$, and $a\neq b$. Multiplying top and bottom of the left hand side by $x$ shows $$\frac{1}{a/x-b/x}=\frac{x}{a-b}$$ and this equals $\frac{1}{a-b}$ if and only if $x=1$.

In short, it can't be true, but my guess is that the book meant to have an $x$ as the numerator of the right hand side of the original equation.

Answer 2

Are you sure it is not $\frac{1}{a/x-b/x}=\frac{x}{a-b}$? Otherwise I agree with you.

Answer 3

$\frac{1}{\frac{a}{x}-\frac{b}{x}} = \frac{1}{\frac{a-b}{x}} = \frac{x}{a-b} \not\equiv \frac{1}{a-b}$

So this is not true in the general case.