document.write( "Question 156955: Please Help.
\n" ); document.write( "(x+12)(x-1)(x+6)>0\r
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Algebra.Com's Answer #115713 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( " $\"%28x%2B12%29%28x-1%29%28x%2B6%29%3E0\"$ Start with the given inequality\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%2B12%29%28x-1%29%28x%2B6%29=0\"$ Set the left side equal to zero\r
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\n" ); document.write( "\n" ); document.write( "Set each individual factor equal to zero:\r
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\n" ); document.write( "\n" ); document.write( " $\"x%2B12=0\"$ , $\"x-1=0\"$ or $\"x%2B6=0\"$ \r
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\n" ); document.write( "\n" ); document.write( "Solve for x in each case:\r
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\n" ); document.write( "\n" ); document.write( " $\"x=-12\"$ , $\"x=1\"$ or $\"x=-6\"$ \r
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\n" ); document.write( "\n" ); document.write( "So our critical values are $\"x=-12\"$ , $\"x=1\"$ and $\"x=-6\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now set up a number line and plot the critical values on the number line\r
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\n" ); document.write( "\n" ); document.write( " $\"number_line%28+600%2C+-13%2C+10%2C-12%2C1%2C-6%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So let's pick some test points that are near the critical values and evaluate them.\r
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\n" ); document.write( "\n" ); document.write( "Let's pick a test value that is less than $\"-12\"$ (notice how it's to the left of the leftmost endpoint):\r
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\n" ); document.write( "\n" ); document.write( "So let's pick $\"x=-13\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%2B12%29%28x-1%29%28x%2B6%29%3E0\"$ Start with the given inequality\r
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\n" ); document.write( "\n" ); document.write( " $\"%28-13%2B12%29%28-13-1%29%28-13%2B6%29%3E+0\"$ Plug in $\"x=-13\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"-98%3E+0\"$ Evaluate and simplify the left side\r
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\n" ); document.write( "\n" ); document.write( "Since the inequality is false, this means that the interval does not work. So this interval is not in our solution set and we can ignore it.\r
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\n" ); document.write( "\n" ); document.write( "Let's pick a test value that is in between $\"-12\"$ and $\"-6\"$ :\r
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\n" ); document.write( "\n" ); document.write( "So let's pick $\"x=-9\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%2B12%29%28x-1%29%28x%2B6%29%3E0\"$ Start with the given inequality\r
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\n" ); document.write( "\n" ); document.write( " $\"%28-9%2B12%29%28-9-1%29%28-9%2B6%29%3E+0\"$ Plug in $\"x=-9\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"90%3E+0\"$ Evaluate and simplify the left side\r
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\n" ); document.write( "\n" ); document.write( "Since the inequality is true, this means that the interval works. So this tells us that this interval is in our solution set.\r
\n" ); document.write( "\n" ); document.write( " So part our solution in interval notation is (

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\n" ); document.write( "\n" ); document.write( "Let's pick a test value that is in between $\"-6\"$ and $\"1\"$ :\r
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\n" ); document.write( "\n" ); document.write( "So let's pick $\"x=-2\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%2B12%29%28x-1%29%28x%2B6%29%3E0\"$ Start with the given inequality\r
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\n" ); document.write( "\n" ); document.write( " $\"%28-2%2B12%29%28-2-1%29%28-2%2B6%29%3E+0\"$ Plug in $\"x=-2\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"-120%3E+0\"$ Evaluate and simplify the left side\r
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\n" ); document.write( "\n" ); document.write( "Since the inequality is false, this means that the interval does not work. So this interval is not in our solution set and we can ignore it.\r
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\n" ); document.write( "\n" ); document.write( "Let's pick a test value that is greater than $\"1\"$ (notice how it's to the right of the rightmost endpoint):\r
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\n" ); document.write( "\n" ); document.write( "So let's pick $\"x=2\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%2B12%29%28x-1%29%28x%2B6%29%3E0\"$ Start with the given inequality\r
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\n" ); document.write( "\n" ); document.write( " $\"%282%2B12%29%282-1%29%282%2B6%29%3E+0\"$ Plug in $\"x=2\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"112%3E+0\"$ Evaluate and simplify the left side\r
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\n" ); document.write( "\n" ); document.write( "Since the inequality is true, this means that the interval works. So this tells us that this interval is in our solution set.\r
\n" ); document.write( "\n" ); document.write( " So part our solution in interval notation is (

)

(

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\n" ); document.write( "\n" ); document.write( "Here's a graph to visually verify the answer\r
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\n" ); document.write( "\n" ); document.write( " $\"+graph%28+500%2C+500%2C+-15%2C+10%2C+-100%2C+100%2C+%28x%2B12%29%28x-1%29%28x%2B6%29%29+\"$
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