document.write( "Question 979701: If $\"+x+=++sqrt%28+pq+%29+\"$ and $\"+y+=+expr%281%2F2%29+%28p%2Bq%29+\"$ prove that x < y \n" ); document.write( "

Algebra.Com's Answer #600996 by Edwin McCravy(20060) $\"\"$ $\"About$

You can put this solution on YOUR website!

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document.write( "Note:  It must be stated that \r\n" );
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document.write( "p ≧ 0, q ≧ 0, pq > 0, p ≠ q, \r\n" );
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document.write( "otherwise the inequality is false.  I'll assume these,\r\n" );
document.write( "but first I'll give two examples to show you that this \r\n" );
document.write( "is false without those assumptions.\r\n" );
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document.write( "It isn't true when p=-8, q=-2 because\r\n" );
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document.write( "Yet 4 < -5 is false.\r\n" );
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document.write( "It also isn't true when p=1, q=1 because\r\n" );
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document.write( "Yet 1 < 1 is false.\r\n" );
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document.write( "It must be stated that p ≧ 0, q ≧ 0, pq > 0, p ≠ q,\r\n" );
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document.write( "It is also easy to see that it is true if p is 0 and q isn't \r\n" );
document.write( "or if q is 0 and p isn't, but they both cannot be 0.\r\n" );
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document.write( "Here is the proof\r\n" );
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document.write( "  assumed as given\r\n" );
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document.write( "The square of any non-zero number is positive, so\r\n" );
document.write( "we can square both sides:\r\n" );
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document.write( "add 4pq to both sides:\r\n" );
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document.write( "Factor the left side\r\n" );
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document.write( "Write left side as the square of a binomial:\r\n" );
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document.write( "Take non-negative square roots of both sides \r\n" );
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document.write( "Multiply both sides by 1/2:\r\n" );
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document.write( "Same as \r\n" );
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document.write( "Edwin

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