Fractional inequality x/(x−1) ≤ 1 − 1/(1−2x)

Text

Solve the following fractional inequality: $\frac{x}{x - 1} \leq 1 - \frac{1}{1-2x}.$

Solution

The right-hand side: $1-\dfrac{1}{1-2x} = \dfrac{(1-2x)-1}{1-2x} = \dfrac{-2x}{1-2x} = \dfrac{2x}{2x-1}$. Move everything to the left over the common denominator $(x-1)(2x-1)$: $\frac{x}{x-1} - \frac{2x}{2x-1} = \frac{x(2x-1) - 2x(x-1)}{(x-1)(2x-1)} = \frac{x}{(x-1)(2x-1)} \le 0.$ Key points: $x=0$ (numerator), $x=\tfrac12$ and $x=1$ (denominator, excluded). Studying the sign of $\dfrac{x}{(x-1)(2x-1)}$:

• $x\le 0$: expression $\le 0$ (with $x=0$ included, giving $0$);
• $\tfrac12 < x < 1$: expression $<0$. $\boxed{x \le 0 \ \vee\ \tfrac12 < x < 1}$