Question

If $\sqrt{4-x}\le 5$ then what is the range of $x$ ?

• A. $x\in [-22,4]$
• B. $x\in [-21,5]$
• C. $x\in [-22,5]$
• D. $x\in [-21,4]$

Answer 1

The correct option is D $x\in [-21,4]$

Given that $\sqrt{4-x}\le 5$
For $c<0:a\ge b\Rightarrow ac\le bc$
Now $\sqrt{4-x}$ exists iff $(4-x)\ge 0$
$\Rightarrow (4-x)-4\ge 0-4$
$\Rightarrow -x\ge (-4)$
$\Rightarrow x\le 4\Rightarrow x\in (-\infty ,4]......\left(i\right)$

Here, $\sqrt{4-x}\ge 0$ and $5>0$
So, on squaring the inequality $\sqrt{4-x}\le 5$, we get
$4-x\le 25$
$\Rightarrow 4-x-4\le 25-4$
$\Rightarrow -x\le 21$
$\Rightarrow x\ge -21$
$\Rightarrow x\in [-21,\infty )......\left(ii\right)$
From (i) and (ii) we get,
$x\in [-21,4]$

Hence option(a) is the correct answer.