Prove: $\Big|\dfrac{1}{x}\Big|=\dfrac{1}{|x|}$

Question

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 $|x|^{-1}$.

Answer 1

Case 1: $x>0$:

$$\left|\frac{1}{x}\right| = \frac{1}{x} = \frac{1}{|x|}$$

Case 2: $x<0$:

$$\left|\frac{1}{x}\right| = -\frac{1}{x} = \frac{1}{-x} = \frac{1}{|x|}$$

Answer 2

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...

$$x*\frac{1}{x} = 1$$

$$Qed$$