Question 1203084
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Show that if |x+3| < 1/2 , then |4x+13| < 3.
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<pre>
Your starting inequality is

    |x+3| < 1/2.     (1)


Taking off the absolute value symbol, it means that

    -1/2 < x+3 < 1/2.      (the compound inequality)


Multiply the last compound inequality by 4  (multiply all three its terms).
You will get an equivalent inequality

    -2 < 4x + 12 < 2.


Add 1 (one) to the last compound inequality (to all its three terms).
You will get an equivalent inequality

    -1 < 4x + 13 < 3.    (2)


But if (2) is valid, then also

    -3 < 4x + 13 < 3     (3)

is valid, too.  


    +--------------------------------------------------------------------------+
    |   In Math, they say "if (2) is valid, then (3) is valid even more so".   |
    +--------------------------------------------------------------------------+


The last inequality (3)  is the same as 

    |4x+13| < 3,

which is requested to prove.


At this point, the proof is complete.
</pre>

Solved.


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Regarding the post by &nbsp;@MathLover1, &nbsp;notice that she solved the given inequality explicitly
(which was not requested), &nbsp;but &nbsp;DID &nbsp;NOT &nbsp;prove the final inequality in full, &nbsp;as it was requested.


So, &nbsp;her post &nbsp;IS &nbsp;NOT &nbsp;the solution to the problem.



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Now, &nbsp;using this example, &nbsp;I would like to educate you a bit on such subjects 
as a mathematical beauty and mathematical elegance.


Notice that in solving this problem, &nbsp;I did not derive an inequality for x explicitly.
I even did not try to do it, &nbsp;because it is &nbsp;NOT &nbsp;NECESSARY &nbsp;for the solution.


I used another way, &nbsp;simply transforming, &nbsp;step by step, &nbsp;the given inequality
to what I needed.


It saved my efforts and was more straightforward.


So, &nbsp;it is mathematically more elegant way comparing with the way with deriving 
the explicit solution for &nbsp;x.


The conception of mathematical elegancy is similar to any other harmony,
which people find in music, &nbsp;in art, &nbsp;in poetry, &nbsp;in prose.


When people unexpectedly find mathematical elegancy, &nbsp;it makes them happy - if
they are really familiar with this feeling of harmony.



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;It is one of the reasons (not a unique), &nbsp;why some people 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;love Math and tend to learn more and more in &nbsp;Math.