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Its old exam im working with:) the problem is solve diffrent so the picture nr 1 at begining is the problem and rest is how u solve it but on third step (the step before last) acording to me it should be wrong:S they wanna get same in the bottom so u multiplicerar with x+1 and x+3 so acording to me it should be (x+1)^2(x+3)^2 can some1 plz explain:)?
Take a simpler example. AddQuote:
Originally Posted by Petrus
You want to get a common denominator, which is 4*13. So
The point to note here is that we only need one of each factor (4 and 13) in the denominator.
So if we have
the common denominator will be (x + 1)(x + 3). We only need one of each factor.
-Dan
in the field of rational functions of x, the quotient:
f(x)/f(x) = 1 (so long as f(x) is not the constant 0-function) (strictly speaking this isn't really true, because we might have to "leave out points where f(x) = 0" whereas the constant function c(x) = 1 is defined for ALL x...but it's true "almost everywhere"...there's only a finite number of x's if we're talking about polynomials where the denominator might be undefined. these points might have to be looked at separately).
so:
see? no squares.
your example is just the same as this with:
f(x) = x - 2
g(x) = x + 1
h(x) = -(x - 1)
k(x) = x + 3
the points -1 and -3 make the one of the two denominators "blow up". at these two values for x, the inequality "makes no sense", on one side of -1, (x - 2)/(x + 1) is large negative (the right side), on the other side (the left side) (x - 2) is large positive, while the fraction (x - 1)/(x + 3) is negative, and a similar consideration holds for x = 3.
so we can only establish the truth of the inequality on these 3 intervals:
(-∞,-3), (-3,-1), and (-1,∞).