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If "x" is positive, "-x" is negative; if "x" is negative, "-x" is positive. Similarly, "x-1" can be positive or negative, depending on the exact value of "x".

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Q: When there is a -x and x -1 is it positive?
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What does a positive number multiplied by a positive number equal?

A positive number multiplied by a positive number will give a positive answer:1 x 1 = 1In contrast, a negative number multiplied by a positive number gives a negative answer:-1 x 1 = -1Finally two negative numbers multiplied together give a positive answer:-1 x -1 = 1


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It is a positive reciprocal. x-a = 1/xa or equivalently, (1/x)a


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Prove that xy add x add y equals 30 has no positive integer solutions?

First we can solve for y by factoring it out: y(x+1)+x=30, so y=(30-x)/(x+1)= -1+31/(x+1). Let's start by assuming x and y are positive integers and look for a logical contradiction. Since y is positive, the right side (30-x)/(x+1) must be positive. Since x (and therefore x+1) is positive, 30-x must be positive. Therefore x is less than 31. But hold on! Since y is an integer, y+1=31/(x+1) is an integer. Since 31 is only divisible by 1 and 31, y+1 is an integer implies that (x+1) is 1 or 31, making x either 0 or 30. However, x is positive and less 30, which is impossible! There it cannot be the case that x and y are both positive integers.AnswerIt's not as complicated as that. Just add 1 to both sides and factorise, and you get: (x+1)(y+1) = 31. Since 31 is prime, one of its factors must be either 1 or -1. So we'd get x+1=1 (so x=0) or else x+1=-1 (so x=-2), or else y=0 or y=-2. But the question says x and y have to be positive.


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