Algebra
# Absolute Value Inequalities

What is the smallest positive integer $n$, such that there exists $n$ real numbers $\{ x_i \} _{i=1}^n$ which satisfy $| x_i | < 1$ and

$| x_1| + |x_2| + |x_3| + \ldots + |x_n| = 165 + |x_1 + x_2 + \ldots + x_n| ?$

As $x$ ranges over all real numbers, what is the smallest value of

$f(x) = |x- 165 | + |x-247 | + |x- 316 | ?$

Solve for $x:$ $\lvert x+16 \rvert - 2\lvert x - 16 \rvert + \lvert x \rvert > 4 .$

As $x$ ranges over all real values, what is the minimum value of $f(x)=|x-333|+|x-631| + |x-856|$?

**Details and assumptions**

The notation $| \cdot |$ denotes the absolute value. The function is given by $|x | = \begin{cases} x & x \geq 0 \\ -x & x < 0 \\ \end{cases}$ For example, $|3| = 3, |-2| = 2$.