Solve 3/2>m-6/5-2m

$$\frac{ 3 }{ 2 } > \frac{ m-6 }{ 5-2m }$$

Solve for m

$$m\in \left(-\infty,\frac{5}{2}\right)\cup \left(\frac{27}{8},\infty\right)$$

Solution Steps

Swap sides so that all variable terms are on the left hand side. This changes the sign direction.

$$\frac{m-6}{5-2m}<\frac{3}{2}$$

Denominator 5-2m cannot be zero since division by zero is not defined. There are two cases.

$$5-2m>0 5-2m<0$$

Consider the case when 5-2m is positive. Move 5 to the right hand side.

$$-2m>-5$$

Divide both sides by -2. Since -2 is negative, the inequality direction is changed.

$$m<\frac{5}{2}$$

The initial inequality does not change the direction when multiplied by 5-2m for 5-2m>0.

$$m-6<\frac{3}{2}\left(5-2m\right)$$

Multiply out the right hand side.

$$m-6<\frac{15}{2}-3m$$

Move the terms containing m to the left hand side and all other terms to the right hand side.

$$m+3m<6+\frac{15}{2}$$

Combine like terms.

$$4m<\frac{27}{2}$$

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.

$$m<\frac{5}{2}$$

Now consider the case when 5-2m is negative. Move 5 to the right hand side.

$$-2m<-5$$

Divide both sides by -2. Since -2 is negative, the inequality direction is changed.

$$m>\frac{5}{2}$$

The initial inequality changes the direction when multiplied by 5-2m for 5-2m<0.

$$m-6>\frac{3}{2}\left(5-2m\right)$$

Multiply out the right hand side.

$$m-6>\frac{15}{2}-3m$$

Move the terms containing m to the left hand side and all other terms to the right hand side.

$$m+3m>6+\frac{15}{2}$$

Combine like terms.

$$4m>\frac{27}{2}$$

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.

$$m>\frac{27}{8}$$

The final solution is the union of the obtained solutions.

$$m\in \left(-\infty,\frac{5}{2}\right)\cup \left(\frac{27}{8},\infty\right)$$