Question 1048119
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"the sum of a number and it's absolute value" translates to the expression {{{x+abs(x)}}} where x is the unknown number.


The question you're asking is: Does {{{x+abs(x)}}} ever equal zero? In other words, are there solutions to {{{x+abs(x) = 0}}}?


Let's find out


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Recall that {{{abs(x) = x}}} if {{{x >= 0}}} or {{{abs(x) = -x}}} if {{{x < 0}}}. So this means that {{{abs(x)}}} breaks down into {{{x}}} or {{{-x}}} depending on what x is.


Let's say {{{x >= 0}}}. That would mean...


{{{x+abs(x)=x+x = 2x}}}


So, {{{x+abs(x)=0}}} turns into {{{2x=0}}} which has the only solution of {{{x = 0}}}


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Now let's say that {{{x < 0}}}. That would mean {{{abs(x) = -x}}}, so,


{{{x+abs(x)=x-x = 0x = 0}}}


So, {{{x+abs(x)=0}}} turns into {{{0=0}}} which has infinitely many solutions. Essentially, you can plug in any negative number in for x into {{{x+abs(x)}}} 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 <font color=red size=5>sometimes</font>


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