document.write( "Question 138675: Solve the inequality $\"3x%5E2%2B13x%2B4%3E0\"$ \n" ); document.write( "

Algebra.Com's Answer #101190 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"3x%5E2%2B13x%2B4%3E0\"$ Start with the given inequality\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%2B4%29%283x%2B1%29%3E0\"$ Factor the left side (note: if you need help with factoring, check out this solver)\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%2B4%29%283x%2B1%29=0\"$ 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( " $\"x%2B4=0\"$ or $\"3x%2B1=0\"$ \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( " $\"x=-4\"$ or $\"x=-1%2F3\"$ \r
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\n" ); document.write( "\n" ); document.write( "So our critical values are $\"x=-4\"$ and $\"x=-1%2F3\"$ \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( " $\"number_line%28+600%2C+-10%2C+10%2C-4%2C-1%2F3%29\"$ \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 $\"-4\"$ (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 $\"x=-5\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%2B4%29%283x%2B1%29%3E0\"$ Start with the given inequality\r
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\n" ); document.write( "\n" ); document.write( " $\"%28-5%2B4%29%283%28-5%29%2B1%29%3E+0\"$ Plug in $\"x=-5\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"14%3E+0\"$ 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 $\"-4\"$ and $\"-1%2F3\"$ :\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( " $\"%28x%2B4%29%283x%2B1%29%3E0\"$ Start with the given inequality\r
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\n" ); document.write( "\n" ); document.write( " $\"%28-2%2B4%29%283%28-2%29%2B1%29%3E+0\"$ Plug in $\"x=-2\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"-10%3E+0\"$ 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 $\"-1%2F3\"$ (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 $\"x=1\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%2B4%29%283x%2B1%29%3E0\"$ Start with the given inequality\r
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\n" ); document.write( "\n" ); document.write( " $\"%281%2B4%29%283%281%29%2B1%29%3E+0\"$ Plug in $\"x=1\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"20%3E+0\"$ 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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