Question 1094630
Using the properties of equality prove that 5&#8730;2 - &#8730;11 < 5&#8730;3 - 3
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EDITED TO:  Using the properties of <em>inequalities</em> prove that 5&#8730;2 - &#8730;11 < 5&#8730;3 - 3
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<pre>
Observe that
{{{ 5sqrt(2) < 5sqrt(3) }}}    <— because {{{ sqrt(2) < sqrt(3) }}}
{{{ sqrt(11) > 3 }}}  <— because {{{ 11 > 3^2 }}}

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Starting with this:
     a < c     ( {{{ 5sqrt(2) < 5sqrt(3) }}} )

we can use the property of inequalities:  if  a < c, then a - b < c - b to write:

     {{{ (5sqrt(2) - sqrt(11)) < (5sqrt(3) - sqrt(11)) }}}   (1)

Now, if  a=b  and c>d then  a-c < b-d, so we can also write:
      {{{ (5sqrt(3) - sqrt(11)) < (5sqrt(3) - 3) }}}    (2)
because {{{ sqrt(11) > 3 }}}
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Combining (1) and (2):
    {{{  (5sqrt(2) - sqrt(11))  <  (5sqrt(3) - sqrt(11))   <  (5sqrt(3) - 3 )}}} 


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Finally, by using  a < b < c  —>  a < c, and applying it to the inequality above:

   {{{  ( 5sqrt(2) - sqrt(11) ) < ( 5sqrt(3) - 3 ) }}}    


[ In words, you are starting with a value less than another (a < c), and then subtracting values from each of 'a' and 'c'.   The value subtracted from 'a' is greater than the value subtracted from 'c' which makes a-b even smaller than c-d.  ]