document.write( "Question 1203084: Show that if |x+3| < 1/2 , then |4x+13| < 3 \n" ); document.write( "

Algebra.Com's Answer #838344 by ikleyn(53874) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "Show that if |x+3| < 1/2 , then |4x+13| < 3.
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document.write( "Your starting inequality is\r\n" );
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document.write( "    |x+3| < 1/2.     (1)\r\n" );
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document.write( "Taking off the absolute value symbol, it means that\r\n" );
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document.write( "    -1/2 < x+3 < 1/2.      (the compound inequality)\r\n" );
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document.write( "Multiply the last compound inequality by 4  (multiply all three its terms).\r\n" );
document.write( "You will get an equivalent inequality\r\n" );
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document.write( "    -2 < 4x + 12 < 2.\r\n" );
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document.write( "Add 1 (one) to the last compound inequality (to all its three terms).\r\n" );
document.write( "You will get an equivalent inequality\r\n" );
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document.write( "    -1 < 4x + 13 < 3.    (2)\r\n" );
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document.write( "But if (2) is valid, then also\r\n" );
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document.write( "    -3 < 4x + 13 < 3     (3)\r\n" );
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document.write( "is valid, too.  \r\n" );
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document.write( "    +--------------------------------------------------------------------------+\r\n" );
document.write( "    |   In Math, they say \"if (2) is valid, then (3) is valid even more so\".   |\r\n" );
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document.write( "The last inequality (3)  is the same as \r\n" );
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document.write( "    |4x+13| < 3,\r\n" );
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document.write( "which is requested to prove.\r\n" );
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document.write( "At this point, the proof is complete.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "Regarding the post by @MathLover1, notice that she solved the given inequality explicitly
\n" ); document.write( "(which was not requested), but DID NOT prove the final inequality in full, as it was requested.\r
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\n" ); document.write( "\n" ); document.write( "So, her post IS NOT the solution to the problem.\r
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\n" ); document.write( "\n" ); document.write( "Now, using this example, I would like to educate you a bit on such subjects
\n" ); document.write( "as a mathematical beauty and mathematical elegance.\r
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\n" ); document.write( "\n" ); document.write( "Notice that in solving this problem, I did not derive an inequality for x explicitly.
\n" ); document.write( "I even did not try to do it, because it is NOT NECESSARY for the solution.\r
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\n" ); document.write( "\n" ); document.write( "I used another way, simply transforming, step by step, the given inequality
\n" ); document.write( "to what I needed.\r
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\n" ); document.write( "\n" ); document.write( "It saved my efforts and was more straightforward.\r
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\n" ); document.write( "\n" ); document.write( "So, it is mathematically more elegant way comparing with the way with deriving
\n" ); document.write( "the explicit solution for x.\r
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\n" ); document.write( "\n" ); document.write( "The conception of mathematical elegancy is similar to any other harmony,
\n" ); document.write( "which people find in music, in art, in poetry, in prose.\r
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\n" ); document.write( "\n" ); document.write( "When people unexpectedly find mathematical elegancy, it makes them happy - if
\n" ); document.write( "they are really familiar with this feeling of harmony.\r
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\n" ); document.write( "\n" ); document.write( " It is one of the reasons (not a unique), why some people
\n" ); document.write( " love Math and tend to learn more and more in Math.\r
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