document.write( "Question 1044149: can you slove this inequality?\r
\n" ); document.write( "\n" ); document.write( "x^3-4x<0 \n" ); document.write( "

Algebra.Com's Answer #659416 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"highlight%28%22Yes+.%22%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Just kidding!
\n" ); document.write( "There are many ways to the answer. The expected way may depend of what you have been learning in class, and teacher preference.
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\n" ); document.write( "1) Just factor that polynomial.
\n" ); document.write( " $\"x%5E3-4x%3C0\"$
\n" ); document.write( " $\"x%28x%5E2-4%29%3C0\"$
\n" ); document.write( " $\"x%28x-2%29%28x%2B2%29%3C0\"$
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\n" ); document.write( "For $\"x=-2\"$ , $\"x=0\"$ , and $\"x=2\"$
\n" ); document.write( " $\"x%5E3-4x=x%28x-2%29%28x%2B2%29=0\"$ .
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\n" ); document.write( "For other values of $\"x\"$ the polynomial is either positive or negative.
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\n" ); document.write( "1a) You could consider the cases
\n" ); document.write( " $\"x%3E2\"$ , $\"0%3Cx%3C2\"$ , $\"-2%3Cx%3C0\"$ , and $\"x%3C-2\"$ individually.
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\n" ); document.write( "1b) You could draw 3 number lines for the factors (one for each factor) and a number line for the product;
\n" ); document.write( "decorate the factor number lines with \"0\" where the factor is zero,
\n" ); document.write( "\"-\" where the factor is negative, and \"+\" where it is positive:
\n" ); document.write( "and then figure out similar decoration for the product number line by multiplying.
\n" ); document.write( "Did your teacher show that way to do it in class?
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\n" ); document.write( "1c) You could realize that
\n" ); document.write( "at each of the zeros the polynomial changes sign (positive to negative or vice versa), and
\n" ); document.write( "for $\"x%3E2\"$ all the factors and $\"x%5E3-4x=x%28x-2%29%28x%2B2%29%3E0\"$ .\r
\n" ); document.write( "\n" ); document.write( "Either way, you realize that the polynomial is negative in the intervals
\n" ); document.write( " $\"%22%28%22\"$ $\"-infinity\"$ $\"%22%2C+-2+%29%22\"$ and $\"%22%28+0+%2C+2+%29%22\"$ ,
\n" ); document.write( "or expressed as an inequality $\"highlight%28x%3C-2%29\"$ and $\"highlight%280%3Cx%3C2%29\"$ .
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\n" ); document.write( "1d) Yet another way is to use \"test numbers\" in those intervals
\n" ); document.write( "Doing that, you see that
\n" ); document.write( "for $\"x=3\"$ , $\"x%5E3-4x=x%28x-2%29%28x%2B2%29=3%2A1%2A5=15%3E0\"$ , so $\"X%3E2\"$ is not the right answer;
\n" ); document.write( "for $\"x=1\"$ , $\"x%5E3-4x=x%28x-2%29%28x%2B2%29=1%2A%28-1%29%2A3=-3%3C0\"$ , so $\"highlight%280%3Cx%3C2%29\"$ works;
\n" ); document.write( "for $\"x=-1\"$ , $\"x%5E3-4x=x%28x-2%29%28x%2B2%29=%28-1%29%2A%28-3%29%2A1=3%3E0\"$ , so $\"-2%3Cx%2C0\"$ is not the right answer, and
\n" ); document.write( "for $\"x=-3\"$ , $\"x%5E3-4x=x%28x-2%29%28x%2B2%29=%28-3%29%2A%28-5%29%2A%28-1%29=-15%3C0\"$ , so $\"highlight%28x%3C-2%29\"$ works.
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\n" ); document.write( "Are there other ways to get to the answer?
\n" ); document.write( "2) Use a graphing calculator?
\n" ); document.write( "3) Use your knowledge (if any) of polynomial function behavior at the ends and at zeros of multiplicity 1. \n" ); document.write( "

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