Practicing Success
Solve $x > \sqrt{(1-x)}$. |
$\left(\frac{\sqrt{5}-1}{2},1\right]$ $\left(\frac{\sqrt{5}+1}{2},1\right]$ $\left[\frac{\sqrt{5}}{2},0\right)$ $\left[0,\frac{\sqrt{5}-1}{2}\right)$ |
$\left(\frac{\sqrt{5}-1}{2},1\right]$ |
Given inequality can be solved by squaring both sides. But sometimes squaring gives extraneous solutions which do not satisfy the original inequality. Before squaring we must restrict x for which terms in the given inequality are well defined. $x > \sqrt{(1-x)}$. Here x must be positive. Now, $\sqrt{1-x}$ is defined only when $1 - x ≥ 0$ or $x ≤ 1$ Thus $0 ≤ x ≤ 1$ Squaring given inequality both sides, $x^2>1-x$ $⇒x^2+x-1>0$ $⇒\left(x-\frac{-1-\sqrt{5}}{2}\right)\left(x-\frac{-1+\sqrt{5}}{2}\right)>0$ $⇒x<\frac{-1-\sqrt{5}}{2}$ or $x>\frac{-1+\sqrt{5}}{2}$ From (1) and (2), $x∈\left(\frac{\sqrt{5}-1}{2},1\right]$ |