Algebra

# Absolute Value: Level 3 Challenges

\begin{aligned} \large |\color{#20A900}x| + \color{#20A900}x + \color{#3D99F6}y &=& \large \color{#624F41}8 \\\large \color{#20A900}x + |\color{#3D99F6}y|-\color{#3D99F6}y &=& \large \color{#69047E}{14} \\ \large \color{#20A900}x + \color{#3D99F6}y &=&\large \ \color{grey}? \end{aligned}

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{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 $|a-d|$?

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