Solve 2-x/x+1geq1

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

Solve for x

$$x\in (-1,\frac{1}{2}]$$

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.

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

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

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

Combine like terms.

$$-2x\geq -1$$

Divide both sides by -2. Since -2 is negative, the inequality direction is changed.

$$x\leq \frac{1}{2}$$

Consider condition x>-1 specified above.

$$x\in (-1,\frac{1}{2}]$$

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.

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

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

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

Combine like terms.

$$-2x\leq -1$$

Divide both sides by -2. Since -2 is negative, the inequality direction is changed.

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

Consider condition x<-1 specified above.

$$x\in \emptyset$$

The final solution is the union of the obtained solutions.

$$x\in (-1,\frac{1}{2}]$$