SOLUTION: I have a lot, but the first one is.... Is the sum of a number and it's absolute value always, sometimes, or never zero????.

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Question 1048119: I have a lot, but the first one is.... Is the sum of a number and it's absolute value always, sometimes, or never zero????.
Answer by jim_thompson5910(35256) (Show Source):
You can put this solution on YOUR website!

"the sum of a number and it's absolute value" translates to the expression where x is the unknown number.

The question you're asking is: Does ever equal zero? In other words, are there solutions to ?

Let's find out

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Recall that if or if . So this means that breaks down into or depending on what x is.

Let's say . That would mean...

So, turns into which has the only solution of

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Now let's say that . That would mean , so,

So, turns into which has infinitely many solutions. Essentially, you can plug in any negative number in for x into and the result will be zero.

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Wrapping things up:

So going back to the question "Is the sum of a number and it's absolute value always, sometimes, or never zero?" the answer would be sometimes

Why sometimes? Because negative values and zero will be solutions while any positive x value will not work.