# Need help with my first college math class - multiple absolute value equation

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?

• 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. – rogerl Oct 20 '18 at 13:25
• So the solutions are 5x = 2018 and 5x = -2018 (this one's wrong it seems.) Thanks. – alcatraz Oct 20 '18 at 13:28
• If x is negative, $|x|=-x$. You get a different equation than your second one in that case. – Paul Oct 20 '18 at 13:39
• Yep, I got it. Ty :) – alcatraz Oct 20 '18 at 13:40

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 $$-x = 2018 \implies x = -2018$$.