Solve x-1/x+1geq2

$$\frac { x - 1 } { x + 1 } \geq 2$$

Solve for x

$$x\in [-3,-1)$$

Solution Steps

Denominator x+1 cannot be zero since division by zero is not defined. There are two cases.

$$x+1>0 x+1<0$$

Consider the case when x+1 is positive. Move 1 to the right hand side.

$$x>-1$$

The initial inequality does not change the direction when multiplied by x+1 for x+1>0.

$$x-1\geq 2\left(x+1\right)$$

Multiply out the right hand side.

$$x-1\geq 2x+2$$

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

$$x-2x\geq 1+2$$

Combine like terms.

$$-x\geq 3$$

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

$$x\leq -3$$

Consider condition x>-1 specified above.

$$x\in \emptyset$$

Now consider the case when x+1 is negative. Move 1 to the right hand side.

$$x<-1$$

The initial inequality changes the direction when multiplied by x+1 for x+1<0.

$$x-1\leq 2\left(x+1\right)$$

Multiply out the right hand side.

$$x-1\leq 2x+2$$

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

$$x-2x\leq 1+2$$

Combine like terms.

$$-x\leq 3$$

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

$$x\geq -3$$

Consider condition x<-1 specified above.

$$x\in [-3,-1)$$

The final solution is the union of the obtained solutions.

$$x\in [-3,-1)$$