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There's an equation that we got assigned to solve in our first college math class. I was alright at math in high school, but I've never seen an absolute value equation similar to this one.

|||||x|+x|+x|+x|+x| = 2018

I'm guessing we have to split it in 2 cases, one in which

+x >= 0

and

-x < 0

How should I approach this problem?

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  • $\begingroup$ Yes, that's right. If $x>0$, then $|x|=x$, right? In which case can you simplify the left-hand side? If $x<0$, then $|x|=-x$, which should again help you simplify the left-hand side. $\endgroup$ – rogerl Oct 20 '18 at 13:25
  • $\begingroup$ So the solutions are 5x = 2018 and 5x = -2018 (this one's wrong it seems.) Thanks. $\endgroup$ – alcatraz Oct 20 '18 at 13:28
  • $\begingroup$ If x is negative, $|x|=-x$. You get a different equation than your second one in that case. $\endgroup$ – Paul Oct 20 '18 at 13:39
  • $\begingroup$ Yep, I got it. Ty :) $\endgroup$ – alcatraz Oct 20 '18 at 13:40
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If $x \geq 0$, then $2x, 3x, 4x, 5x$ are also non-negative, and the LHS is $$ |||||x|+x|+x|+x|+x| = ||||2x|+x|+x|+x| = \dotsb = 5x,$$ so $5x = 2018 \implies x = 2018/5$.

If instead $x < 0$, then $|x| = -x$, and the LHS is $$ |||||x|+x|+x|+x|+x| = ||||-x+x|+x|+x|+x| = ||||0|+x|+x|+x| = |||x|+x|+x| = ||-x+x|+x| = |0+x| = |x| = -x,$$ so $-x = 2018 \implies x = -2018$.

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  • $\begingroup$ I did get the first one right. Mind explaining the second one? $\endgroup$ – alcatraz Oct 20 '18 at 13:31
  • $\begingroup$ Thanks! I got it. $\endgroup$ – alcatraz Oct 20 '18 at 13:39

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