Algebra
# Absolute Value Inequalities

\[\left | 1 - \frac{|x|}{1 + |x|} \right| \geq \frac{1}{2} \]

The solution set to this inequality is \(a \le x \le b.\) What is \(b-a?\)

How many integers \(x\) satisfy the inequality \( |x-2000|+|x| \leq 9999\)?

How many ordered pairs of integers \( (x,y)\) are there that satisfy \( |x| + |y| \leq 10 \)?

Consider all monic polynomials \(f(x) = x^2 + bx + c \), where \(b\) and \(c\) are real numbers. What is the minimum value of \(N\), where

\[ N = \max_{x \in [-10,10]} \vert f(x) \vert? \]

**Details and assumptions**

The last equation states: "The maximum value of the absolute value of \(f(x) \) , as \(x\) ranges from \(-10\) to \(10\) inclusive".

\[\large 2=|x-2|+|x-4|\]

Find the solution set of the above equation.

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