Algebra

# 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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