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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.

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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