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

Text

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

Solution

Move everything to the left over the common denominator $x(2-x)$: $\frac{2(2-x) - x\cdot x - x(2-x)}{x(2-x)} = \frac{4 - 4x}{x(2-x)} \ge 0 \;\Longleftrightarrow\; \frac{1-x}{x(2-x)} \ge 0.$ Key points: $x=1$ (numerator, included), $x=0,\ x=2$ (denominator, excluded). Sign study of $\dfrac{1-x}{x(2-x)}$:

• $0 < x \le 1$: positive (with $x=1$ included, giving $0$);
• $x > 2$: positive. $\boxed{0 < x \le 1 \ \vee\ x > 2}$