document.write( "Question 1018501: How on earth do you prove that:
\n" ); document.write( "|x|=|-x|\r
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\n" ); document.write( "\n" ); document.write( "|x|≥0\r
\n" ); document.write( "\n" ); document.write( "Just seems so hard to prove it? \n" ); document.write( "

Algebra.Com's Answer #634589 by ikleyn(52803) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "How on earth do you prove that:
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document.write( "1.  Let x be a non-negative real number: x >= 0.\r\n" );
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document.write( "    Then |x| = x.\r\n" );
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document.write( "    Also then -x is negative number, and |-x| = -(-x) by the definition of the modulus. So, |-x| = -(-x) = x.\r\n" );
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document.write( "    Thus we proved that if x >= 0, then  |x| = |-x|.\r\n" );
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document.write( "2.  Now consider the case, when x is negative real number: x < 0.\r\n" );
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document.write( "    Then |x| = -x by the definition of the modulus.\r\n" );
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document.write( "    Also then -x is positive number, and hence |-x| = -x.\r\n" );
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document.write( "    Thus we proved that if x < 0, then  |x| = |-x|.\r\n" );
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document.write( "Thus we proved that in all cases |x| = |-x|.\r\n" );
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document.write( "QED.\r\n" );
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