Algebra

Absolute Value

Absolute Value: Level 3 Challenges

         

x+x+y=8x+yy=14x+y= ?\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 x2+x6=0 \left| x \right|^2 + \left| x \right| - 6 = 0 .

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

The set of solutions of the equation x1+11=x+11+1 \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

ab=ab. |a-b| = a-b.

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

ab=ab? |a-b| = a-b?

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

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

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