Solve 4x-2/x+2<4

$$\frac{ 4x-2 }{ x+2 } < 4$$

Solve for x

$$x>-2$$

Solution Steps

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

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

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

$$x>-2$$

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

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

Multiply out the right hand side.

$$4x-2<4x+8$$

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

$$4x-4x<2+8$$

Combine like terms.

$$0<10$$

Consider condition x>-2 specified above.

$$x>-2$$

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

$$x<-2$$

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

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

Multiply out the right hand side.

$$4x-2>4x+8$$

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

$$4x-4x>2+8$$

Combine like terms.

$$0>10$$

Consider condition x<-2 specified above.

$$x\in \emptyset$$

The final solution is the union of the obtained solutions.

$$x>-2$$