zaarathelab wrote:If y is an integer and y=|x|+x, is y=0?
1) x<0
2) y<1
when would y be zero? If |x|+x=0, then x=-|x|. Since an absolute value is ≥0, this means that y will equal zero if x is negative (or zero).
another way to look at the same is that for |x|+x to equal zero, the x has to be negative to counter the "positive" of |x|.
This is why stat. (1) is sufficient: if x<0, then y=0 for any value of x.
Stat. (2): y is an integer, so if it's smaller than 1, it has to be either zero or negative. we've already seen that if x is negative, y is equal to zero. Can we find a negative y? No way: for y to be negative, something needs to "drag" |x|+x below the zero line. Since |x| is non-negative, x has to be negative for that to happen - and we've already seen that a negative x still only gets y=0. So stat. (2) is sufficient as well.
Note that if the stem did not say y is an integer, we could use x=1/4 to find y=|1/4|+1/4 =1/2 to be smaller than 1 but not zero. The statement is only sufficient because y has to be an integer <1: zero or negative.
