pure maths - inequality

In AL Pure Maths, it is typical to ask you to prove a given statement of inequality. Normally the approach is NOT to prove from nothing (at starting point) to the statement (as end point), as it is virtually impossible to do so.
The clue is:
On the rough work paper, work from the given statement (as your starting point), de-composing it back to the supposed-to-be starting simple inequalities. And then, write your rough work in its "opposite manner" onto your answer book.

Start from the answer, and then expand both sides of the inequality. Then '砌' that inequality back to the formula that you know (e.g. AM>=GM, etc......)

This is naural because you have not learned many useful inequality such as AM≧GM and 1/(1-x) > 1+x . If you still want to try, the following may be helpful:

1. Observing that is there any symmetry of unknowns?
2. If there is a quadratic inequality, try to use the discriminant to obtain a better inequality.
3. Try to split the positive, negative and the zero part(s).
4. Should M.I. be applied?

I am a pure maths student who get A in CE maths and a. maths .
You should think out the steps from the answer. It seems thst it is out of logic, but it looks like you are playing a maze, walking from the exit back to the entrance is always easier than vice versa. It is impossible for us to know how,and where to start from the normal direction .
Let us consider a typical example:
Note: iff means if and only if
prove that
(2k-1)/4k<=root(3k+2/3k+1), given k>=1
In reversed dir,
squaring them,we get
iff(2k-1/4k)^2<=(3k-2)/(3k+1)
iff(2k-1)^2(3k+1)<=4k^2(3k-2)
iff12k^3-8k^2-k+1<=12k^3-8k
iff-k+1<=0
iff k>=1