Solve for $x$ the following system of inequalities: $|x-1| \leq 5$ and $|x| \geq 2$.
(A) Given the inequalities:
$|x-1| \leq 5$ $(i)$
$|x| \geq 2$ $(ii)$
Solving $(i)$:
$|x-1| \leq 5$
$-5 \leq x-1 \leq 5$
$-5+1 \leq x \leq 5+1$
$-4 \leq x \leq 6$
So,$x \in [-4, 6]$.
Solving $(ii)$:
$|x| \geq 2$
$x \leq -2$ or $x \geq 2$
So,$x \in (-\infty, -2] \cup [2, \infty)$.
Combining the solutions from $(i)$ and $(ii)$:
We look for the intersection of $[-4, 6]$ and $(-\infty, -2] \cup [2, \infty)$.
Intersection $= [-4, -2] \cup [2, 6]$.
$|x-1| \leq 5$ $(i)$
$|x| \geq 2$ $(ii)$
Solving $(i)$:
$|x-1| \leq 5$
$-5 \leq x-1 \leq 5$
$-5+1 \leq x \leq 5+1$
$-4 \leq x \leq 6$
So,$x \in [-4, 6]$.
Solving $(ii)$:
$|x| \geq 2$
$x \leq -2$ or $x \geq 2$
So,$x \in (-\infty, -2] \cup [2, \infty)$.
Combining the solutions from $(i)$ and $(ii)$:
We look for the intersection of $[-4, 6]$ and $(-\infty, -2] \cup [2, \infty)$.
Intersection $= [-4, -2] \cup [2, 6]$.
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