Very confused and having trouble solving this inequality!?!?

Question

F(x)=|1+x|/|1-x^2|*times* (x^2-5x+6)(x-1)^3/
(4-x^2)(x+3)(2x+1)^2 <- less than or equal to 0
hope you can kinda of understand the formatting! Any help would be greatly appreciated!

Answer 1

From what I can see, you have:

F(x) = A(x) B(x) where:

A(x) = |1+x|/|1-x^2|
B(x) = P(x)/Q(x)
P(x) = (x^2-5x+6)(x-1)^3
Q(x) = (4-x^2)(x+3)(2x+1)^2

and you are trying to determine when F(x) <= 0

As it happens, A(x) can never be negative, so while you do have to worry about it being 0 (though it can never be 0), most of your effort has to be on where B(x) is negative, and where it ceases to exist because Q(x) is 0.

Both P(x) and Q(x) are polynomials and so are continuous. So the first step is to find all their roots and put them in increasing numerical order. (This is easy because most of the factoring has been done for you)

Once you get all the roots, you need to work through each interval in turn to determine the sign of F over that interval.

Just to make your life easier, remember that:
a^2 - b^2 = (a-b)(a+b)

This applies to the (1 - x^2) term in A(x) and to the (4 - x^2) term in Q(x)

Answer 2

x-a million / 2 + a million > x + a million = > simplify x-a million/ 3 > x + a million => multiply by potential of three x-a million > 3 (x + a million) => distribute x-a million > 3x + 3 => simplify and remedy for x x > 3x + 4 => subtract x 0 > 2x + 4 => subtract 4 2x < - 4 => divide by potential of two x < -2