document.write( "Question 135271: $\"x%2B%2810%29%2F%28x-2%29=%28x%5E2%2B3x%29%2F%28x-2%29\"$ \n" ); document.write( "

Algebra.Com's Answer #99088 by solver91311(24713) $\"\"$ $\"About$

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$\"x%2B%2810%29%2F%28x-2%29=%28x%5E2%2B3x%29%2F%28x-2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Put everything on the left:
\n" ); document.write( " $\"x%2B%2810%29%2F%28x-2%29-%28%28x%5E2%2B3x%29%2F%28x-2%29%29=0\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"x-2\"$ is the common denominator:
\n" ); document.write( " $\"%28x%28x-2%29%29%2F%28x-2%29%2B%2810%29%2F%28x-2%29-%28%28x%5E2%2B3x%29%2F%28x-2%29%29=0\"$ \r
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\n" ); document.write( "\n" ); document.write( "Expand and collect like terms:
\n" ); document.write( " $\"%28x%5E2-2x%2B10-x%5E2-3x%29%2F%28x-2%29=0\"$
\n" ); document.write( " $\"%28-5x%2B10%29%2F%28x-2%29=0\"$ \r
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\n" ); document.write( "\n" ); document.write( "Note that $\"a%2Fb=0\"$ if and only if $\"a=0\"$ AND $\"b%3C%3E0\"$ . So set the numerator equal to 0 and solve:\r
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\n" ); document.write( "\n" ); document.write( " $\"-5x%2B10=0\"$
\n" ); document.write( " $\"-5x=-10\"$
\n" ); document.write( " $\"x=2\"$ \r
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\n" ); document.write( "\n" ); document.write( "However, if $\"x=2\"$ , then the denominator $\"x-2\"$ equals 0, so $\"x=2\"$ must be excluded as a root of this equation, and there are no other roots. \r
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\n" ); document.write( "\n" ); document.write( "Therefore the solution set of $\"x%2B%2810%29%2F%28x-2%29=%28x%5E2%2B3x%29%2F%28x-2%29\"$ is the empty set.
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