Question

Find a value for x if 1/x > x

Answer 1

Since x can't be zero (it sits at the denominator of the left hand side), let's split cases:

If x is positive

It this case, we can multiply both sides by x without changing the inequality sign:

$\dfrac{1}{x} > x \iff 1 > x^2 \iff x \in (-1,1)$

But since we're assuming that x is positive, we can only accept the numbers between 0 and 1.

If x is negative

It this case, in order to multiply both sides by x we have to change the inequality sign:

$\dfrac{1}{x} > x \iff 1 < x^2 \iff x \in (-\infty,-1) \cup (1,\infty)$

But since we're assuming that x is negative, we can only accept the numbers which are smaller than -1.

So, putting all the pieces together, we have that the solution is given by the following interval:

$(-\infty, -1) \cup (0,1)$