document.write( "Question 149234: Solve the inequality (x + 3)(x + 1)(x – 5) < 0 and write the solution set both in interval notation and in set notation. Show work/explanation. \n" ); document.write( "
Algebra.Com's Answer #109482 by jim_thompson5910(35256)\"\" \"About 
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\n" ); document.write( "\"%28x%2B3%29%28x%2B1%29%28x-5%29%3C0\" Start with the given inequality\r
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\n" ); document.write( "\n" ); document.write( "\"%28x%2B3%29%28x%2B1%29%28x-5%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%2B3=0\", \"x%2B1=0\" or \"x-5=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=-3\", \"x=-1\" or \"x=5\"\r
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\n" ); document.write( "\n" ); document.write( "So our critical values are \"x=-3\", \"x=-1\" and \"x=5\"\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-3%2C-1%2C5%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 \"-3\" (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=-4\"\r
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\n" ); document.write( "\n" ); document.write( "\"%28x%2B3%29%28x%2B1%29%28x-5%29%3C0\" Start with the given inequality\r
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\n" ); document.write( "\n" ); document.write( "\"%28-4%2B3%29%28-4%2B1%29%28-4-5%29%3C+0\" Plug in \"x=-4\"\r
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\n" ); document.write( "\n" ); document.write( "\"-27%3C+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 ()\r
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\n" ); document.write( "\n" ); document.write( "Let's pick a test value that is in between \"-3\" 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%2B3%29%28x%2B1%29%28x-5%29%3C0\" Start with the given inequality\r
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\n" ); document.write( "\n" ); document.write( "\"%28-2%2B3%29%28-2%2B1%29%28-2-5%29%3C+0\" Plug in \"x=-2\"\r
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\n" ); document.write( "\n" ); document.write( "\"7%3C+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 \"-1\" and \"5\":\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%2B3%29%28x%2B1%29%28x-5%29%3C0\" Start with the given inequality\r
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\n" ); document.write( "\n" ); document.write( "\"%282%2B3%29%282%2B1%29%282-5%29%3C+0\" Plug in \"x=2\"\r
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\n" ); document.write( "\n" ); document.write( "\"-45%3C+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 ()\r
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\n" ); document.write( "\n" ); document.write( "Let's pick a test value that is greater than \"5\" (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=6\"\r
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\n" ); document.write( "\n" ); document.write( "\"%28x%2B3%29%28x%2B1%29%28x-5%29%3C0\" Start with the given inequality\r
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\n" ); document.write( "\n" ); document.write( "\"%286%2B3%29%286%2B1%29%286-5%29%3C+0\" Plug in \"x=6\"\r
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\n" ); document.write( "\n" ); document.write( "\"63%3C+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( "Summary:\r
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\n" ); document.write( "\n" ); document.write( "So the solution in interval notation is:\r
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\n" ); document.write( "\n" ); document.write( "() ()\r
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\n" ); document.write( "\n" ); document.write( "Also, the answer in set notation is\r
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\n" ); document.write( "\n" ); document.write( "Here's a graph to visually verify our answer:\r
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\n" ); document.write( "\n" ); document.write( "\"+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+%28x%2B3%29%28x%2B1%29%28x-5%29%29+\" Graph of \"y=%28x%2B3%29%28x%2B1%29%28x-5%29\"
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