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Show that if |x+3| < 1/2 , then |4x+13| < 3.
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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.
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| 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.
Solved.
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Regarding the post by @MathLover1, notice that she solved the given inequality explicitly
(which was not requested), but DID NOT prove the final inequality in full, as it was requested.
So, her post IS NOT the solution to the problem.
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Now, using this example, I would like to educate you a bit on such subjects
as a mathematical beauty and mathematical elegance.
Notice that in solving this problem, I did not derive an inequality for x explicitly.
I even did not try to do it, because it is NOT NECESSARY for the solution.
I used another way, simply transforming, step by step, the given inequality
to what I needed.
It saved my efforts and was more straightforward.
So, it is mathematically more elegant way comparing with the way with deriving
the explicit solution for x.
The conception of mathematical elegancy is similar to any other harmony,
which people find in music, in art, in poetry, in prose.
When people unexpectedly find mathematical elegancy, it makes them happy - if
they are really familiar with this feeling of harmony.
It is one of the reasons (not a unique), why some people
love Math and tend to learn more and more in Math.