document.write( "Question 1025840: For what values of x is
\n" ); document.write( "{x^2 + x + 3}/{2x^2 + x - 6} >= 0?\r
\n" ); document.write( "\n" ); document.write( "Note: Be thorough and explain why all points in your answer are solutions and why all points outside your answer are not solutions. \n" ); document.write( "

Algebra.Com's Answer #641285 by Fombitz(32388) $\"\"$ $\"About$

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Find the critical points of the functions.
\n" ); document.write( " $\"x%5E2%2Bx%2B3=0\"$
\n" ); document.write( "The roots of this quadratic are complex so no real roots.
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\n" ); document.write( " $\"2x%5E2%2Bx-6=0\"$
\n" ); document.write( " $\"%28x%2B2%29%282x-3%29=0\"$
\n" ); document.write( " $\"x%2B2=0\"$
\n" ); document.write( " $\"x=-2\"$
\n" ); document.write( "and
\n" ); document.write( " $\"2x-3=0\"$
\n" ); document.write( " $\"2x=3\"$
\n" ); document.write( " $\"x=3%2F2\"$
\n" ); document.write( "Break up the number line using the two critical points.
\n" ); document.write( "Region 1 : ( $\"-infinity\"$ , $\"-2\"$ )
\n" ); document.write( "Region 2 : ( $\"-2\"$ , $\"3%2F2\"$ )
\n" ); document.write( "Region 3 : ( $\"3%2F2\"$ , $\"infinity\"$ )
\n" ); document.write( "In each region, choose a point (not an endpoint) check the quotient.
\n" ); document.write( "Region 1 : $\"x=-3\"$
\n" ); document.write( " $\"%28%28-3%29%5E2-3%2B3%29%2F%282%28-3%29%5E2-3-6%29=%289%29%2F%289%29=1\"$
\n" ); document.write( "Region 2 : $\"x=0\"$
\n" ); document.write( " $\"%28%280%29%5E2%2B0%2B3%29%2F%282%280%29%5E2%2B0-6%29=%283%29%2F%28-6%29=-1%2F2\"$
\n" ); document.write( "Region 3 : $\"x=2\"$
\n" ); document.write( " $\"%28%282%29%5E2%2B2%2B3%29%2F%282%282%29%5E2%2B2-6%29=%289%29%2F%284%29\"$
\n" ); document.write( "So when,
\n" ); document.write( "( $\"-infinity+%3Cx%3C+-2\"$ ) , $\"f%28x%29%3E0\"$
\n" ); document.write( "( $\"-2+%3Cx%3C+3%2F2\"$ ) , $\"f%28x%29%3C0\"$
\n" ); document.write( "( $\"3%2F2+%3Cx%3C+infinity\"$ ) , $\"f%28x%29%3E0\"$
\n" ); document.write( "The quotient is never equal to zero since the numerator is never equal to zero.
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