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# Absolute Value

Absolute value is a mathematician's way of judging numbers by their magnitude rather than their positive/negative value. It is the distance of the number from 0 on a number line.

# Absolute Value Inequalities

Which of the following represents the correct value of $$x$$ if $\lvert 8x \rvert + b > c,$ and $$c>b$$?

Solve for $$x$$:

$x^2 - 2x -3 < 3 |x-1|.$

If the solution set of the system of inequalities: $\begin{cases}\lvert a+1 \rvert < 2\\ \lvert b-1 \rvert <12 \end{cases}$ is $$x < a+b < y,$$ what are $$x$$ and $$y?$$

For all non-zero real numbers $$(x, y)$$, which of the following is larger:

$A = \frac{ |x| } { 1 + |x| } + \frac{ |y| } { 1 + |y| } \text{ or } B = \frac{ |x+y| } { 1 + |x+y| } ?$

The inequality $$\lvert 2x+8 \rvert + 16 < 32$$ implies that $$a < x < b$$. What is $$b-a$$?

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