Algebra

# Absolute Value: Level 3 Challenges

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

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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