# Evaluating Absolute Value Expression Within Ranges 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.

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.
• 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 $[-3,4)$? – jessica Oct 12 '13 at 19:35
• 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??? – jessica Oct 12 '13 at 20:12