solving inequalities and induction help?

Question

Hey could anyone help me with these 2 questions they are from last years exam so im assuming ill need to know how to do this. thanks in advance

http://img512.imageshack.us/img512/4853/mathsagainhardfc5.jpg

Answer 1

2 / 2x - 3 ≤ 4

2x - 3(2 / 2x - 3 ≤ 4(2x - 3)

2 ≤ 8x - 12

2 + 12 ≤ 8x - 12 + 12

14 ≤ 8x

14 / 8 ≤ 8x / 8

14 / 8 ≤ x

1 6/8 ≤ x

1 3/4 ≤ x

1.75 ≤ x

- - - - - - - - -s-

Answer 2

x >= 7/4 is NOT the only solution to the inequality.
This is because when you multiply an equality the sign is reversed if the thing you are multiplying by is negative.
Thus the problem should be split into two parts
2 =< 8x - 12 IF 2x - 3 > 0 leading to x >= 7/4
AND 2 >= 8x - 12 IF 2x - 3 < 0 leading to x < 3/2. (Of course
x = 3/2 is not allowed otherwise the LHS is undefined.)
You can easily check that this alternative is correct by putting
x = 0 into the inquality and it works.
So final answer is x >= 7/4 or x < 3/2.

Answer 3

2/(2x-3) <_ 4 ?

2 <_ 4(2x-3)

2 <_ 8x - 12

14 <_ 8x

1.75 <_ x

That's about it :)