Solve 1-x/x+2<-1

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

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.

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

Multiply out the right hand side.

$$1-x<-x-2$$

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

$$-x+x<-1-2$$

Combine like terms.

$$0<-3$$

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.

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

Multiply out the right hand side.

$$1-x>-x-2$$

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

$$-x+x>-1-2$$

Combine like terms.

$$0>-3$$

Consider condition x<-2 specified above.

$$x<-2$$

The final solution is the union of the obtained solutions.

$$x<-2$$