Solve 4x/2x+3>2

$$\frac{ 4x }{ 2x+3 } > 2$$

Solve for x

$$x<-\frac{3}{2}$$

Solution Steps

Denominator 2x+3 cannot be zero since division by zero is not defined. There are two cases.

$$2x+3>0 2x+3<0$$

Consider the case when 2x+3 is positive. Move 3 to the right hand side.

$$2x>-3$$

Divide both sides by 2. Since 2 is positive, the inequality direction remains the same.

$$x>-\frac{3}{2}$$

The initial inequality does not change the direction when multiplied by 2x+3 for 2x+3>0.

$$4x>2\left(2x+3\right)$$

Multiply out the right hand side.

$$4x>4x+6$$

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

$$4x-4x>6$$

Combine like terms.

$$0>6$$

Consider condition x>-\frac{3}{2} specified above.

$$x\in \emptyset$$

Now consider the case when 2x+3 is negative. Move 3 to the right hand side.

$$2x<-3$$

Divide both sides by 2. Since 2 is positive, the inequality direction remains the same.

$$x<-\frac{3}{2}$$

The initial inequality changes the direction when multiplied by 2x+3 for 2x+3<0.

$$4x<2\left(2x+3\right)$$

Multiply out the right hand side.

$$4x<4x+6$$

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

$$4x-4x<6$$

Combine like terms.

$$0<6$$

Consider condition x<-\frac{3}{2} specified above.

$$x<-\frac{3}{2}$$

The final solution is the union of the obtained solutions.

$$x<-\frac{3}{2}$$