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September 11th, 2009, 04:39 AM   #1
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solving (2x+1)/(2x-1)>=1

Hi All,

Just trying to solve:

(2x+1)/(2x-1)>=1

I try:
2x+1>=2x-1
0>=-2 (not the answer I want)

I tried:
(2x+1)(2x-1)/(2x-1)(2x-1)>=1
4x^2-1>=4x^2-4x+1
-2>=-4x
x>=1/2 (can't be this, it is undefined)

I also tried:
(2x+1)(2x+1)/(2x-1)(2x+1)>=1
4x^2+4x+1>=4x^2-1
4x+1>=-1
4x>=-2
x>=-1/2 (but this answer doesnt work with the inequality...0>=1??)

Not sure what to do now...by looking at the graph I think x>=1/2...but like i said, this is undefined...
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September 11th, 2009, 03:58 PM   #2
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Re: solving (2x+1)/(2x-1)>=1

You have to consider 2 cases separately x>1/2 and x<1/2. When multiplying by 2x-1 you will get 1 > -1 in the first case (true) and 1 < -1 in the second case (false). So the answer you are expecting x > 1/2 is correct.
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September 12th, 2009, 04:01 PM   #3
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Re: solving (2x+1)/(2x-1)>=1

is that really the answer, even though it is undefined at x=1/2
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September 13th, 2009, 03:24 AM   #4
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Quote:
Originally Posted by solarscott
Just trying to solve:

(2x+1)/(2x-1)>=1

I try: 2x+1>=2x-1
When solving rational inequalities, you can not multiply through by the denominator!

Why? Because you do not know what the sign on the denominator is (it's a variable expression!), so you can't know whether or not to flip the inequality sign!

Instead, move everything over to one side of the inequality:



Convert to a common denominator:







Now find the x-values which make for zeroes of the numerator and undefined values for the denominator, use these x-values to split up the number line, and then find the intervals on which the rational expression is positive....
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September 13th, 2009, 06:28 PM   #5
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Clearly, 2/(2x - 1) ? 0 ? (2x - 1)/2 > 0 (since equality isn't possible in the leftmost inequality),
i.e., x - 1/2 > 0, i.e., x > 1/2.
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