Question

The solution of the inequality |x - 1| < 2is​

Answer 1

Solution:

Given Inequality:

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

There are two possible cases:

$\rm 1. \: \: x - 1 < 2 \: \: iff \: \: x - 1 > 0$

$\rm 2. \: \: - (x - 1) < 2 \: \: iff \: \: x - 1 < 0$

Solving (i), we get:

$\rm \longrightarrow x - 1 < 2$

$\rm \longrightarrow x - 1 + 1< 2 + 1$

$\rm \longrightarrow x < 3$

Solving (ii), we get:

$\rm \longrightarrow - (x - 1) < 2$

$\rm \longrightarrow x - 1 > -2$

$\rm \longrightarrow x > -1$

Combining both, we get:

$\rm \longrightarrow -1 < x < 3$

$\rm \longrightarrow x \in ( - 1,3)$

★ Which is our required answer.

Answer:

$\rm \hookrightarrow x \in ( - 1,3)$

Note:

Absolute function is defined as:

$\rm \longrightarrow |x| = \begin{cases} \rm x \: \: iff \: \: x \geqslant 0 \\ \rm - x \: \: iff \: \: x < 0 \end{cases}$