Express the function in piecewise form without using absolute values.

Question

y = |x - 1| + |2 - x|

Answer 1

First, rewrite it as y=|x-1|+|x-2| since |x-2|=|2-x|. Next watch the following video: The key to the problem is to find the interval's where the function changes. Those intervals depend on when the expression inside the absolute values equal 0. So, for this problem, x=1 and x=2. Thus the relevant intervals are (-inf, 1), (1, 2) and (2, inf) where inf=infinite. In each interval we evaluate whether the expression inside of the absolute value is negative or positive for x values in that interval. Interval 1: (-inf, 1) both expressions are negative so we write: y = -(x-1)-(x-2) = - 2x+3 Interval 2: (1, 2) the first expression is positive, the second is still negative so we write: y = (x-1)-(x-2) = 1 Interval 3: (2, inf) both expressions are positive, thus: y = (x-1)+(x-2) = 2x -3 Here are some other examples of how to do this and use it to solve absolute value equations:

Answer 2

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