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If \(y = |2 + x| - |2 - x|\) and \(|2x - 15| < 2,\) how many integer values can \(y\) take?

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Re: If \(y = |2 + x| - |2 - x|\) and \(|2x - 15| < 2,\) how many integer values can \(y\) take?

by regor60 » Sun Aug 22, 2021 10:48 am
Start with the last statement first.

Assume 2X-15 is positive and <2.

Any X >8 will make expression >2.

X=8 makes expression=1 which works. X<8 will make expression negative, which doesn't work.

So X=8 in this case.

Now assume 2X-15 is negative, in which case removing from absolute value requires rearranging to 15-2X.

15-2X<2 and positive. Any X >7 makes expression negative which doesn't work. Any X < 7 makes result >2, which doesn't work, so X=7.

So now we have two cases to use in the first two expressions, X=7 and X=8.

Y= |2+8| - |2-8|= 10-6= 4.
and Y= |2+7| -|2-7|= 9-5=4

Answer1,B
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