SOLUTION: If {{{5<a<7}}}{{{""<b<14}}},then prove that {{{5/14<a/b<1}}}

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Question 215084:
If 5%3Ca%3C7%22%22%3Cb%3C14,then prove that 5%2F14%3Ca%2Fb%3C1
Answer by Edwin McCravy(20067) About Me  (Show Source):
You can put this solution on YOUR website!
If 5%3Ca%3C7%22%22%3Cb%3C14 then prove that 5%2F14%3Ca%2Fb%3C1

b%3C14 is given

Multiply both sides by 1%2F%2814b%29 which is positive and
does not reverse inequality:

b%281%2F%2814b%29%29%3C14%2F%2814b%29

cross%28b%29%281%2F%2814cross%28b%29%29%29%3Ccross%2814%29%2F%28cross%2814%29b%29

1%2F14+%3C+1%2Fb

Multiply both sides by a which is positive and does
not reverse the inequality:

(1)       a%2F14%3Ca%2Fb

5%3Ca is given

Multiply both sides by 1%2F14

(2)      5%2F14%3Ca%2F14

By inequalities (1) and (2),

5%2F14%3Ca%2F14%3Ca%2Fb

So 

(3)  5%2F14%3Ca%2Fb

Now we only need show that a%2Fb%3C1

We are given that a%3C7%3Cb so

a%3Cb

Multiply both sides by 1%2Fb

a%2Fb%3Cb%2Fb

(4)  a%2Fb%3C1

By (3) and (4),

5%2F14+%3C+a%2Fb+%3C+1

which was to be proved.

Edwin