Solve the following inequalities for x:

$\frac{1}{x} < 1$

$(−∞,0)∪(1,∞)$

The correct answer is Option (3) → $(−∞,0)∪(1,∞)$

Given $\frac{1}{x}< 1$. First, we note that $x≠ 0$.

Since $x^2 > 0$ for all $x ∈ R, x ≠0,$

$\frac{1}{x}<1 ⇒ \frac{1}{x}.x^2 < x^2$   (multiplying by $x^2$)

$⇒ x < x^2 ⇒ 0 < x^2-x$

$⇒ x(x-1)>0$   ...(1)

Mark the numbers 0 and 1 on the real line.

By the method of intervals, the inequality (1) is satisfied when $x > 1$ or $x < 0$.

∴ The solution set is $(−∞,0)∪(1,∞)$.