document.write( "Question 1087301: Solve the inequality. Write your answer in interval notation. \r
\n" ); document.write( "\n" ); document.write( "(x + 1)/((x + 5)(x - 2)) < or = 0 \n" ); document.write( "

Algebra.Com's Answer #701610 by Fombitz(32388) $\"\"$ $\"About$

You can put this solution on YOUR website!
Break up the number line into regions using the critical values of the function,
\n" ); document.write( " $\"x=-5\"$
\n" ); document.write( " $\"x=-1\"$
\n" ); document.write( " $\"x=2\"$
\n" ); document.write( "So the four regions would be,
\n" ); document.write( "Region 1: $\"x%3C-5\"$
\n" ); document.write( "Region 2: $\"-5%3Cx%3C-1\"$
\n" ); document.write( "Region 3: $\"-1%3Cx%3C2\"$
\n" ); document.write( "Region 4: $\"x%3E2\"$
\n" ); document.write( "For each region pick a point in the region (not an endpoint) and test the inequality.
\n" ); document.write( "If true, it's part of the solution set.
\n" ); document.write( "If false, it's not part of the solution set.
\n" ); document.write( "The value is not important but whether it's negative or positive.
\n" ); document.write( "Region 1: $\"x=-6\"$
\n" ); document.write( " $\"%28x+%2B+1%29%2F%28%28x+%2B+5%29%28x+-+2%29%29+%3C=0\"$
\n" ); document.write( " $\"%28-6%2B+1%29%2F%28%28-6+%2B+5%29%28-6+-+2%29%29+%3C=0\"$
\n" ); document.write( " $\"%28-5%29%2F%28%28-1%29%28-8%29%29%3C=0\"$
\n" ); document.write( "True
\n" ); document.write( "Region 2: $\"x=-2\"$
\n" ); document.write( " $\"%28x+%2B+1%29%2F%28%28x+%2B+5%29%28x+-+2%29%29+%3C=0\"$
\n" ); document.write( " $\"%28-2%2B+1%29%2F%28%28-2+%2B+5%29%28-2+-+2%29%29+%3C=0\"$
\n" ); document.write( " $\"%28-1%29%2F%28%283%29%28-4%29%29%3C=0\"$
\n" ); document.write( "False
\n" ); document.write( "Region 3: $\"x=0\"$
\n" ); document.write( " $\"%28x+%2B+1%29%2F%28%28x+%2B+5%29%28x+-+2%29%29+%3C=0\"$
\n" ); document.write( " $\"%281%29%2F%28%285%29%28-2%29%29+%3C=0\"$
\n" ); document.write( "True
\n" ); document.write( "Region 4: $\"x=3\"$
\n" ); document.write( " $\"%28x+%2B+1%29%2F%28%28x+%2B+5%29%28x+-+2%29%29+%3C=0\"$
\n" ); document.write( " $\"%283%2B+1%29%2F%28%283+%2B+5%29%283+-+2%29%29+%3C=0\"$
\n" ); document.write( " $\"%284%29%2F%28%288%29%281%29%29%3C=0\"$
\n" ); document.write( "False
\n" ); document.write( "So then,
\n" ); document.write( "( $\"-infinity\"$ , $\"-5\"$ )U( $\"-1\"$ , $\"2\"$ )
\n" ); document.write( "Now dealing with the endpoints.
\n" ); document.write( "Since the inequality includes an equal sign, it's allowed as long as it doesn't make the denominator equal to zero.
\n" ); document.write( "In this case, $\"x=-1\"$ is allowed but $\"x=-5\"$ and $\"x=2\"$ are not allowed.
\n" ); document.write( "Designate inclusion with brackets and exclusion with parentheses.
\n" ); document.write( "( $\"-infinity\"$ , $\"-5\"$ )U[ $\"-1\"$ , $\"2\"$ )
\n" ); document.write( ".
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\n" ); document.write( "The illustration shows the regions and the solution set (where the function lies on or below the x-axis)
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