Question

The set of values of $x$ for which the inequality
$\sqrt{x-5}>\sqrt{7-x}$ holds true is

• A. $(6,7]$
• B. $(6,\infty )$
• C. $[6,\infty )$
• D. $[6,7]$

Answer 1

The correct option is A

$(6,7]$

Given
$\sqrt{x-5}>\sqrt{7-x}$

For expression to be defined

$x-5\ge 0\Rightarrow x\ge 5\cdots \left(1\right)$

$7-x\ge 0\Rightarrow x\le 7\cdots \left(2\right)$

Also $\sqrt{x-5}>\sqrt{7-x}$

Squaring both sides

$\Rightarrow x-5>7-x$

$\Rightarrow 2x>12$

$\Rightarrow x>6\cdots \left(2\right)$

From inequations $\left(1\right)$, $\left(2\right)$ and $\left(3\right)$,
We get,
$x\in (6,7]$

Hence the correct answer is D.