Algebra
# Absolute Value

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 $\left| n + S + P \right|$?

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

$|a-b| = a-b?$

$\begin{aligned} |a - b | &=& 2 \\ |b - c | &=& 3 \\ |c - d | &=& 4 \\ \end{aligned}$

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 $|a-d|$?