Solve x-4/x+2geq2

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

Solve for x

$$x\in [-8,-2)$$

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.

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

Multiply out the right hand side.

$$x-4\geq 2x+4$$

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

$$x-2x\geq 4+4$$

Combine like terms.

$$-x\geq 8$$

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

$$x\leq -8$$

Consider condition x>-2 specified above.

$$x\in \emptyset$$

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.

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

Multiply out the right hand side.

$$x-4\leq 2x+4$$

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

$$x-2x\leq 4+4$$

Combine like terms.

$$-x\leq 8$$

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

$$x\geq -8$$

Consider condition x<-2 specified above.

$$x\in [-8,-2)$$

The final solution is the union of the obtained solutions.

$$x\in [-8,-2)$$