Solving Inequalities: 0<(2x+1)(x-2) or 0>(2x+1)(x-2)?

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The discussion focuses on solving the inequality (4x-4)/(x+2)< 2x-3, which simplifies to analyzing the product (2x+1)(x-2) under different conditions based on the sign of x+2. Participants confirm the initial steps taken and discuss how to determine the sign of the product by finding its zeros and testing intervals. It is noted that the intervals to consider are based on critical points, including -2, and that checking points within these intervals helps identify where the inequality holds true. The conclusion drawn is that x>2 is a potential solution, but it is pointed out that the interval -2<x<-1/2 also needs to be considered. The conversation emphasizes the importance of interval testing in solving inequalities.
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Im trying to solve (4x-4)/(x+2)< 2x-3

I get it down to 0<(2x+1)(x-2) if x+2>0

0>(2x+1)(x-2) if x+2<0

Are these right so far? I am not sure what to do now with the product being bigger or smaller than 0. Thanks for your time.
 
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deryk said:
Im trying to solve (4x-4)/(x+2)< 2x-3

I get it down to 0<(2x+1)(x-2) if x+2>0

0>(2x+1)(x-2) if x+2<0

Are these right so far? I am not sure what to do now with the product being bigger or smaller than 0. Thanks for your time.
Right so far.
You have products
the sign of a product depends on the sighns of the factors
ab>0
means
a>0 and b>0
or
a>0 and b<0
ab<0
means
a>0 and b<0
or
a<0 and b>0

another way to think about it is (2x+1)(x-2) is a continuos function
find out where the zeros are
call them a and b with a<b
consider the intervals (since -2 is also an important number)
x<-2
-2<x<a
a<x<b
b<x
all the points in one of these intervals satisfy the inequality or none do
so checking one point in each intervals tells you if the whole interval satifies the inequality
 
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thanks lurf lurf .I got x>2. Does anyone know if that's right?
 
deryk said:
thanks lurf lurf .I got x>2. Does anyone know if that's right?
you missed -2<x<-1/2
consider for example x=-1
(4x-4)/(x+2)< 2x-3
(4(-1)-4)/((-1)+2)< 2(-1)-3
(-4-4)/1<-2-3
-8<-5
 
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