Question

If $||x-2|-5|\le$ 10, then

• A. -20 $\le$ x $\le$ 15
• B. -13 $\le$ x $\le$ 17
• C. -5 $\le$ x $\le$ 27
• D. -5 $\le$ x $\le$ 17

Answer 1

The correct option is B

-13 $\le$ x $\le$ 17

We will remove the modulus step by step. First the outer one and then the inner modulus.

Let $|x-2|$ = y

$|y-5|$ $\le$ 10

$\Rightarrow$ -10 $\le$ y - 5 $\le$ 10

$\Rightarrow$ -5 $\le$ y $\le$ 15 (adding 5 throughout)

But y =$|x-2|$ |,modulus can't be negative y $\ge$ 0

$\Rightarrow$ 0 $\le$ y $\le$ 15 (Because the minimum value it can take is 0)

$\Rightarrow$ 0 $\le$ $|x-2|$ $\le$ 15

$|x-2|$ $\ge$ 0 for all x

So the only condition is

$|x-2|$ $\le$ 15

$\Rightarrow$ -15 $\le$ x - 2 $\le$ 15

$\Rightarrow$ -13 $\le$ x $\le$ 17