Solve x-2/x+4<2

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

Solve for x

$$x\in \left(-\infty,-10\right)\cup \left(-4,\infty\right)$$

Solution Steps

Denominator x+4 cannot be zero since division by zero is not defined. There are two cases.

$$x+4>0 x+4<0$$

Consider the case when x+4 is positive. Move 4 to the right hand side.

$$x>-4$$

The initial inequality does not change the direction when multiplied by x+4 for x+4>0.

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

Multiply out the right hand side.

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

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

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

Combine like terms.

$$-x<10$$

Divide both sides by -1. Since -1 is negative, the inequality direction is changed.

$$x>-10$$

Consider condition x>-4 specified above.

$$x>-4$$

Now consider the case when x+4 is negative. Move 4 to the right hand side.

$$x<-4$$

The initial inequality changes the direction when multiplied by x+4 for x+4<0.

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

Multiply out the right hand side.

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

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

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

Combine like terms.

$$-x>10$$

Divide both sides by -1. Since -1 is negative, the inequality direction is changed.

$$x<-10$$

Consider condition x<-4 specified above. The result remains the same.

$$x<-10$$

The final solution is the union of the obtained solutions.

$$x\in \left(-\infty,-10\right)\cup \left(-4,\infty\right)$$