TITLE: Prove: $\Big|\dfrac{1}{x}\Big|=\dfrac{1}{|x|}$
QUESTION [0 upvotes]: Can you help me to prove:
$$\Big|\dfrac{1}{x}\Big|=\dfrac{1}{|x|},$$ if $x\neq 0$.
I think the best way is recalling the definition of $|x|^{-1}$.

REPLY [1 votes]: Multiply both sides by |x| you will get the following:
$$|x|*|\frac{1}{x}| = 1$$
Now we can safely remove the absolute mark since we know that on the left side we will always have a positive number either $x$ is positive $(+ * + = +)$ or negative $(- * - = +)$ and now
on the left side is a multiplication of a number and it's reciprocal which always equals 1...
$$x*\frac{1}{x} = 1$$
$$Qed$$
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