Question

Solve for x ∈R if | x – 1 | > 5.​

Answer 1

Solution:

Given Inequality:

$\rm \longrightarrow |x - 1| > 5,x \in R$

Now, there can be two possible cases:

$\rm \longrightarrow \begin{cases} \rm x - 1 > 5 \: \: iff \: \: x - 1 > 0 \\ \rm - (x - 1) >5 \: \: iff \: x - 1 < 0\end{cases}$

Solving (i), we get:

$\rm \longrightarrow x - 1 > 5$

$\rm \longrightarrow x - 1 +1 > 5 + 1$

$\rm \longrightarrow x > 6$

$\rm \longrightarrow x\in (6, \infty )$

Solving (ii), we get:

$\rm \longrightarrow - (x - 1) > 5$

$\rm \longrightarrow (x - 1) < - 5$

$\rm \longrightarrow x -1 + 1 < - 5 + 1$

$\rm \longrightarrow x < - 4$

$\rm \longrightarrow x\in ( - \infty , - 4)$

Combining both, we get:

$\rm \longrightarrow x\in ( - \infty , - 4) \: \cup \:(6, \infty )$

★ Which is our required answer.

Answer:

$\rm \hookrightarrow x\in ( - \infty , - 4) \: \cup \:(6, \infty )$

Note:

Absolute function is defined as:

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