Solving an Inequality: 1/x < 1/(x-1)

Solve the inequality \( \frac{1}{x} < \frac{1}{x-1} \).

To find the solution set for the inequality, combine the fractions and simplify. Consider the critical points where the denominator equals zero and analyze the intervals on a number line.

Move \( \frac{1}{x-1} \) to the left: \( \frac{1}{x} – \frac{1}{x-1} < 0 \).

Combine the fractions on the left: \( \frac{(x-1) – x}{x(x-1)} < 0 \).

Simplify the numerator: \( x – x + 1 = 1 \Rightarrow -\frac{1}{x(x-1)} < 0 \).

The denominator is zero when x=0 or x=1. Make a number line to check the signs:

-∞ <—— 0 ——|—— 1 ——> +∞

Test points in each interval to determine the truth value of the inequality.

Therefore, the solution set is \( \{x \mid x < 0 \text{ or } x > 1\} \).