Question 215084
If {{{5<a<7}}}{{{""<b<14}}} then prove that {{{5/14<a/b<1}}}
<pre><font size = 4 color = "indigo"><b>
{{{b<14}}} is given

Multiply both sides by {{{1/(14b)}}} which is positive and
does not reverse inequality:

{{{b(1/(14b))<14/(14b)}}}

{{{cross(b)(1/(14cross(b)))<cross(14)/(cross(14)b)}}}

{{{1/14 < 1/b}}}

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

(1)       {{{a/14<a/b}}}

{{{5<a}}} is given

Multiply both sides by {{{1/14}}}

(2)      {{{5/14<a/14}}}

By inequalities (1) and (2),

{{{5/14<a/14<a/b}}}

So 

(3)  {{{5/14<a/b}}}

Now we only need show that {{{a/b<1}}}

We are given that {{{a<7<b}}} so

{{{a<b}}}

Multiply both sides by {{{1/b}}}

{{{a/b<b/b}}}

(4)  {{{a/b<1}}}

By (3) and (4),

{{{5/14 < a/b < 1}}}

which was to be proved.

Edwin</pre>