Question

Inqualitites solve this please​

Answer 1

We're given the inequality,

$\longrightarrow ||x-1|-2|<5$

$\longrightarrow ||x-1|-2|\in(-\infty,\ 5)$

Since $|x|\in[0,\ \infty)\quad\forall x\in\mathbb{Z},$

$\longrightarrow ||x-1|-2|\in(-\infty,\ 5)\cap[0,\ \infty)$

$\longrightarrow ||x-1|-2|\in[0,\ 5)$

On removing the outer modulus, we get,

$\longrightarrow |x-1|-2\in(-5,\ 5)$

because $|x|\in[0,\ a)\,\iff\,x\in(-a,\ a).$

$\longrightarrow |x-1|\in(-5+2,\ 5+2)$

$\longrightarrow |x-1|\in(-3,\ 7)$

As we said earlier, since $|x|\in[0,\ \infty)\quad\forall x\in\mathbb{Z},$

$\longrightarrow |x-1|\in(-3,\ 7)\cap[0,\ \infty)$

$\longrightarrow |x-1|\in[0,\ 7)$

On removing the modulus,

$\longrightarrow x-1\in(-7,\ 7)$

$\longrightarrow x\in(-7+1,\ 7+1)$

$\longrightarrow\underline{\underline{x\in(-6,\ 8)}}$

This is the solution of the inequality.