Question

If $|x+5|\ge 10$, then

• A. $x\in (-15,5]$
• B. $x\in (-5,5]$
• C. $x\in (-\infty ,-15]\cup [5,\infty )$
• D. $x\in [-\infty ,-15)\cup (5,\infty )$

Answer 1

The correct option is C $x\in (-\infty ,-15]\cup [5,\infty )$

$|x+5|\ge 10\cdots \left(A\right)$
Case 1: $x+5<0,\text{i.e.}x<-5$
Then, $|x+5|=-x-5$
Then, $\left(A\right)\text{becomes}-x-5\ge 10$
$\Rightarrow x\le -15$
But $x<-5\therefore x\le -15$
$\Rightarrow x\in (-\infty ,-15]\cdots \left(i\right)$

Case 2: $x+5\ge 0,\text{i.e.}x\ge -5$
Then, $|x+5|=x+5$
Then, $\left(A\right)\text{becomes}x+5\ge 10$
$\Rightarrow x\ge 5$
But $x\ge -5\therefore x\ge 5$
$\Rightarrow x\in [5,\infty )\cdots \left(ii\right)$

From $\left(i\right)and\left(ii\right)$
$x\in (-\infty ,-15]\cup [5,\infty )$