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 March 17th, 2011, 08:28 AM #1 Newbie   Joined: Mar 2011 Posts: 2 Thanks: 0 Inequality I'm having trouble with this inequality: 1/(x-5) < 4/5x I'm pretty sure the first step is to do this: 1/(x-5) - 4/5x < 0 But I'm stuck here. Could anyone nudge me in the right direction?
 March 17th, 2011, 08:44 AM #2 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,948 Thanks: 1140 Math Focus: Elementary mathematics and beyond Re: Inequality $\frac{1}{x\,-\,5}\,\lt\,\frac{4}{5x}$ $\frac{1}{x\,-\,5}\,-\,\frac{4}{5x}\,\lt\,0$ $\frac{x\,+\,20}{5x(x\,-\,5)}\,\lt\,0$
 March 17th, 2011, 09:29 AM #3 Newbie   Joined: Mar 2011 Posts: 2 Thanks: 0 Re: Inequality Thank you!
March 18th, 2011, 01:39 AM   #4
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Quote:
 Originally Posted by greg1313 $\frac{x\,+\,20}{5x(x\,-\,5)}\,\lt\,0$
x ? 0, x ? 5
x + 20 > 0, 5x(x - 5) < 0 or x + 20 < 0, 5x(x - 5) > 0
For 5x(x - 5) < 0: 5x > 0, x - 5 < 0 or 5x < 0, x - 5 > 0
x > 0, x < -5 or x < 0, x > -5
For 5x(x - 5) > 0: 5x > 0, x - 5 > 0 or 5x < 0, x - 5 < 0
x > 0, x > -5 or x < 0, x < -5
For x + 20 > 0 and x + 20 < 0: x > -20 and x < -20, respectively
x > -20, x > 0, x < -5 or x < 0, x > -5 OR x < -20, x > 0, x > -5 or x < 0, x < -5
x < 0, x > -5 is equivalent to -5 < x < 0
x > 0, x > -5 is equivalent to x > 0
x < 0, x < -5 is equivalent to x < -5
x > -20, x > 0, x < -5 or -5 < x < 0 OR x < -20, x > 0 or x < -5
x > -20 and -5 < x < 0 is equivalent to -5 < x < 0
x = 1 does not satisfy x > 0, x < -5
-5 < x < 0 OR x < -20, x > 0 or x < -5
x = 1 does not satisfy x < -20 and x > 0
x < -20 and x < -5 is equivalent to x < -20
? -5 < x < 0 or x < -20

March 18th, 2011, 03:07 AM   #5
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Re:

Quote:
Originally Posted by johnny
Quote:
 Originally Posted by greg1313 $\frac{x\,+\,20}{5x(x\,-\,5)}\,\lt\,0$
x ? 0, x ? 5
x + 20 > 0, 5x(x - 5) < 0 or x + 20 < 0, 5x(x - 5) > 0
For 5x(x - 5) < 0: 5x > 0, x - 5 < 0 or 5x < 0, x - 5 > 0
x > 0, x < -5 or x < 0, x > -5
For 5x(x - 5) > 0: 5x > 0, x - 5 > 0 or 5x < 0, x - 5 < 0
x > 0, x > -5 or x < 0, x < -5
For x + 20 > 0 and x + 20 < 0: x > -20 and x < -20, respectively
x > -20, x > 0, x < -5 or x < 0, x > -5 OR x < -20, x > 0, x > -5 or x < 0, x < -5
x < 0, x > -5 is equivalent to -5 < x < 0
x > 0, x > -5 is equivalent to x > 0
x < 0, x < -5 is equivalent to x < -5
x > -20, x > 0, x < -5 or -5 < x < 0 OR x < -20, x > 0 or x < -5
x > -20 and -5 < x < 0 is equivalent to -5 < x < 0
x = 1 does not satisfy x > 0, x < -5
-5 < x < 0 OR x < -20, x > 0 or x < -5
x = 1 does not satisfy x < -20 and x > 0
x < -20 and x < -5 is equivalent to x < -20
? -5 < x < 0 or x < -20
i think it would have been easier to multiply
the whole equation by the square of the denominator
giving u:
$5x(x-5)(x+20)$
then just draw it and determine negative domains

 March 18th, 2011, 03:22 AM #6 Senior Member   Joined: Apr 2007 Posts: 2,140 Thanks: 0 Your solution is much easier. Thanks, daniel_an!
 March 18th, 2011, 04:30 AM #7 Newbie   Joined: Mar 2011 Posts: 4 Thanks: 0 Re: Inequality Wow! Nice solution.

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