# Solve absolute value inequality. |x-1|\geq 2

Solve absolute value inequality.
$|x-1|\ge 2$
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scoollato7o
Step 1
the given inequality is:
$|x-1|\ge 2$
we have to solve the given inequality.
Step 2
the given inequality is $|x-1|\ge 2$
as we know that if $|x|\ge a$
then
that implies $x\in \left(-\mathrm{\infty },-a\right]\cup \left[a,\mathrm{\infty }\right)$
therefore,
$|x-1|\ge 2$
then

$x-1\le -2$
$x\le -2+1$
$x\le -1$
or
$x-1\ge 2$
$x\ge 2+1$
$x\ge 3$
Step 3
therefore the solution of the given inequality is
that implies the solution of the given inequality is $x\in \left(-\mathrm{\infty },-1\right]\cup \left[3,\mathrm{\infty }\right)$

Donald Cheek
Step 1
The absolute value inequality is given as,
$|x-1|\ge 2$
If an absolute inequality is given as, $|f\left(x\right)|\ge a$,
$|f\left(x\right)|\ge a$
$=\left\{\begin{array}{ll}f\left(x\right)\ge a& f\left(x\right)>0\\ f\left(x\right)\le -a& f\left(x\right)<0\end{array}$
Step 2
On solving the inequality, we get
$x-1\ge 2$
$x\ge 2+1$
$x\ge 3$...(i)
And,
$x-1\le -2$
$x\le -2+1$
$x\le -1$...(ii)
Therefore, the interval for the solution of the given inequality is $x\in \left(-\mathrm{\infty },-1\right]\cup \left[3,\mathrm{\infty }\right)$