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, \( |3| = 3, |-2| = 2 \).

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