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 Rubashov March 17th, 2011 08:28 AM

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?

 greg1313 March 17th, 2011 08:44 AM

Re: Inequality

$\frac{1}{x\,-\,5}\,\lt\,\frac{4}{5x}$

$\frac{1}{x\,-\,5}\,-\,\frac{4}{5x}\,\lt\,0$

$\frac{x\,+\,20}{5x(x\,-\,5)}\,\lt\,0$

 Rubashov March 17th, 2011 09:29 AM

Re: Inequality

Thank you!

 johnny March 18th, 2011 01:39 AM

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

 daniel_an March 18th, 2011 03:07 AM

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

 johnny March 18th, 2011 03:22 AM

Your solution is much easier. Thanks, daniel_an!

 julieta March 18th, 2011 04:30 AM

Re: Inequality

Wow! Nice solution.

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