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\frac{m-6}{5-2m}<\frac{3}{2}
Swap sides so that all variable terms are on the left hand side. This changes the sign direction.
5-2m>0 5-2m<0
Denominator 5-2m cannot be zero since division by zero is not defined. There are two cases.
-2m>-5
Consider the case when 5-2m is positive. Move 5 to the right hand side.
m<\frac{5}{2}
Divide both sides by -2. Since -2 is negative, the inequality direction is changed.
m-6<\frac{3}{2}\left(5-2m\right)
The initial inequality does not change the direction when multiplied by 5-2m for 5-2m>0.
m-6<\frac{15}{2}-3m
Multiply out the right hand side.
m+3m<6+\frac{15}{2}
Move the terms containing m to the left hand side and all other terms to the right hand side.
4m<\frac{27}{2}
Combine like terms.
m<\frac{27}{8}
Divide both sides by 4. Since 4 is positive, the inequality direction remains the same.
m<\frac{5}{2}
Consider condition m<\frac{5}{2} specified above.
-2m<-5
Now consider the case when 5-2m is negative. Move 5 to the right hand side.
m>\frac{5}{2}
Divide both sides by -2. Since -2 is negative, the inequality direction is changed.
m-6>\frac{3}{2}\left(5-2m\right)
The initial inequality changes the direction when multiplied by 5-2m for 5-2m<0.
m-6>\frac{15}{2}-3m
Multiply out the right hand side.
m+3m>6+\frac{15}{2}
Move the terms containing m to the left hand side and all other terms to the right hand side.
4m>\frac{27}{2}
Combine like terms.
m>\frac{27}{8}
Divide both sides by 4. Since 4 is positive, the inequality direction remains the same.
m>\frac{27}{8}
Consider condition m>\frac{5}{2} specified above. The result remains the same.
m\in \left(-\infty,\frac{5}{2}\right)\cup \left(\frac{27}{8},\infty\right)
The final solution is the union of the obtained solutions.