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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.

Level 3

         

\[\begin{eqnarray} \large |\color{green}x| + \color{green}x + \color{blue}y &=& \large \color{brown}8 \\\large \color{green}x + |\color{blue}y|-\color{blue}y &=& \large \color{purple}{14} \\ \large \color{green}x + \color{blue}y &=&\large \ \color{grey}? \end{eqnarray} \]

Consider the equation \( \left| x \right|^2 + \left| x \right| - 6 = 0 \).

Let \( n \) be the number of real roots, \( S \) be the sum of those roots, and \( P \) be the product of those roots. What is \(\left| n + S + P \right|\)?

The set of solutions of the equation \[ \bigg\lvert \Big\lvert \lvert x-1 \rvert + 1 \Big\rvert - 1 \bigg\rvert = \bigg\lvert \Big\lvert \lvert x+1 \rvert - 1 \Big\rvert + 1 \bigg\rvert\] is a disjoint union of one or more segments. Find the sum of their lengths.

Lazy Liz doesn't like absolute values notation, and often drops them from her equations. She always writes

\[ |a-b| = a-b. \]

How many of the \( 11 \times 11 \) ordered pairs of integers \( (a, b) \), each of which are between 0 and 10 inclusive, are there, such that

\[ |a-b| = a-b? \]

\[ \begin{eqnarray} |a - b | &=& 2 \\ |b - c | &=& 3 \\ |c - d | &=& 4 \\ \end{eqnarray} \]

Given that \(a,b,c,d\) are real numbers that satisfy the system of equations above, what is the sum of all distinct values of \(|a-d| \)?

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