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I am trying to evaluate an absolute value expression but I am struggling to know whether to place a (+) or a (-) on each expression when evaluating each interval. For example, is there a quick method that can take a second to realize whether $|4-x|$ gets a $+/-$ sign in the interval $[-3,4)$? (Without having to say $|4-x|=4-x$ if $4-x>0$ or $4>x$). I say this because on my math exams, time is constrained and I need to evaluate long absolute expression in seconds.

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In order to get to to know if |4−x| gets a +/− sign in the interval [−3,4) just plug in the end points of your interval.
For example out here plug in -3 and 3.99 for $x$ and since the result is positive in both cases $$|4-x|=4-x$$ If you have [5,6] as your interval then you plug in 5 and 6 for $x$ and since the result is negative in both cases: $$|4-x|=x-4$$ If you have [3,6] as your interval then you plug in 3 and 6 for $x$ and since the result is negative in one case and positive in the other split the interval into two-[3,4] and [4,6] to get a better idea.

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  • $\begingroup$ Without plugging numbers. Is there a way to quickly look at the interval and the root (in this case 4) and find out whether $|4-x|$ gets a +/- in the interval $[-3,4)$? $\endgroup$ – jessica Oct 12 '13 at 19:35
  • $\begingroup$ However if you have something like |(x-4)(x-3)(x-2)(x-1)| it would be good if you could plot it on the number line and then evaluate a +/-. $\endgroup$ – iajnr Oct 12 '13 at 19:40
  • $\begingroup$ So for the interval [-8,3), $|x+3|$ is negative (-) in this interval. $|x+3|$ has a root of $-3$ and the interval [-8,3) is to the left of the root. Same thought process can be used for $|x+8|$ which has a root of -8. The interval [-8,3) is to the right of the root, so it gets a (+). However for $|4-x|$ the root is 4 and the interval [-8,3) is to the left of the root but the expression gets a (+). Why??? $\endgroup$ – jessica Oct 12 '13 at 20:12
  • $\begingroup$ Oh sorry for that advice:I guess plugging in is the best option and graphing the function is equally good. $\endgroup$ – iajnr Oct 12 '13 at 20:18

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