document.write( "Question 147170This question is from textbook College Algebra
\n" ); document.write( ": Sovle each inequality by making a sign graph of the factors. State the solution set in iterval notation and graph it?\r
\n" ); document.write( "\n" ); document.write( "5x-x^2<4\r
\n" ); document.write( "\n" ); document.write( "Can you please help me sovle this problem? \n" ); document.write( "

Algebra.Com's Answer #107502 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"5x-x%5E2%3C4\"$ Start with the given inequality.\r
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\n" ); document.write( "\n" ); document.write( " $\"5x-x%5E2-4%3C0\"$ Subtract 4 from both sides.\r
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\n" ); document.write( "\n" ); document.write( " $\"-x%5E2%2B5x-4%3C0\"$ Rearrange the terms.\r
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\n" ); document.write( "\n" ); document.write( " $\"-%28x-1%29%28x-4%29%3C0\"$ Factor the left side\r
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\n" ); document.write( "\n" ); document.write( " $\"-%28x-1%29%28x-4%29=0\"$ 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( " $\"x-1=0\"$ or $\"x-4=0\"$ \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( " $\"x=1\"$ or $\"x=4\"$ \r
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\n" ); document.write( "\n" ); document.write( "So our critical values are $\"x=1\"$ and $\"x=4\"$ \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( " $\"number_line%28+600%2C+-10%2C+10%2C1%2C4%29\"$ \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 $\"1\"$ (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 $\"x=0\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"-x%5E2%2B5x-4%3C0\"$ Start with the given inequality\r
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\n" ); document.write( "\n" ); document.write( " $\"-%280-1%29%280-4%29%3C+0\"$ Plug in $\"x=0\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"-4%3C+0\"$ 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 (

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\n" ); document.write( "\n" ); document.write( "Let's pick a test value that is in between $\"1\"$ and $\"4\"$ :\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( " $\"-x%5E2%2B5x-4%3C0\"$ Start with the given inequality\r
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\n" ); document.write( "\n" ); document.write( " $\"-%282-1%29%282-4%29%3C+0\"$ Plug in $\"x=2\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"2%3C+0\"$ 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 greater than $\"4\"$ (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 $\"x=5\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"-x%5E2%2B5x-4%3C0\"$ Start with the given inequality\r
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\n" ); document.write( "\n" ); document.write( " $\"-%285-1%29%285-4%29%3C+0\"$ Plug in $\"x=5\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"-4%3C+0\"$ 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 (

)

(

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\n" ); document.write( "\n" ); document.write( "Here is the graph of the solution set:\r
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\n" ); document.write( "\n" ); document.write( "Note:\r
\n" ); document.write( "\n" ); document.write( "There is an open circle at $\"x=1\"$ which means that we're excluding that value from the solution set.\r
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\n" ); document.write( "\n" ); document.write( "Also, there is an open circle at $\"x=4\"$ which means that we're excluding that value from the solution set.
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