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

[deryk]

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.

 

[lurflurf]

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

 

[deryk]

thanks lurf lurf .I got x>2. Does anyone know if that's right?

 

[lurflurf]

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