stuck on solving inequality's for math!!!?

Question

the three im stuck on are
(x-2)(3-x)>0
x^4-1<0
(x-2)/(x-3)>or equals to 0

PLEASE HELP!!!

Answer 1

Let's look at your first problem this way: a product of the factors x - 2 and 3 - x is positive if and only if both factors have the same sign. Therefore, either

(1) x - 2 > 0 and 3 - x > 0 , or

(2) x - 2 < 0 and 3 - x < 0 .

(1) requires that x > 2 AND 3 > x; that is, 2 < x < 3. On the other hand, (2) requires that x < 2 AND 3 < x. Clearly it is not possible that x is less than 2 and greater than 3 at the same time, so (2) cannot be satisfied.

The only solution is 2 < x < 3.

Your second problem can be analyzed by factoring x^4 - 1. We find x^4 - 1 = (x + 1)*(x - 1)*(x^2 + 1) < 0. The product of three factors is negative if ond only if (3) all three factors are negative or (4) two are positive and one is negative. Clearly, case (3) is not possible since x^2 + 1 is never negative. Case (4) requires three subcases:
(a) x + 1 < 0 and the other two are positive, (b) x - 1 < 0 and others are > 0, and
(c) x^2 + 1 < 0 (which is impossible).

Hope you got the idea.

Answer 2

3>x>2 for the first one
x>1^1/4 for the second one
x>3 for the third one

Answer 3

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