Question

Solve for x:x > $\sqrt{1-x}$

• A.
• B. U
• C.
• D. [1,)

Answer 1

The correct option is C

We may think of simple squaring & getting the values of x like shown below

x > $\sqrt{1-x}$

Squaring,

$x^2$ > 1 - x

$x^2$ - x - 1 > 0 $\Rightarrow$ x ∈ $(-\infty ,\frac{-1-\sqrt{5}}{2})$ U $(\frac{-1+\sqrt{5}}{2},\infty )$------(1)

But sometimes, squaring gives extra solution which don't satisfy the original inequality.

For eg, at x = 2 $\Rightarrow$ 2 > $\sqrt{1-2}$

$\Rightarrow$ 2 > $\sqrt{-1}$ (not valid)

So, first we carefully see possible values of x from original inequality x > $\sqrt{1-x}$

Square root always gives $\Rightarrow$ $\sqrt{1-x}$ ≥ 0

So, x > $\sqrt{1-x}$ $\Rightarrow$ x is always positive ------------(2)

Also in $\sqrt{1-x}$, 1-x ≥ 0

1 ≥ x

$\Rightarrow$ x ≤ 1--------------(3)

Intersection of (1), (2) and (3) gives

x ∈ $(\frac{\sqrt{5}-1}{2},1]$