Solve x+2/x+7<2

$$\frac { x + 2 } { x + 7 } < 2$$

Solve for x

$$x\in \left(-\infty,-12\right)\cup \left(-7,\infty\right)$$

Solution Steps

Denominator x+7 cannot be zero since division by zero is not defined. There are two cases.

$$x+7>0 x+7<0$$

Consider the case when x+7 is positive. Move 7 to the right hand side.

$$x>-7$$

The initial inequality does not change the direction when multiplied by x+7 for x+7>0.

$$x+2<2\left(x+7\right)$$

Multiply out the right hand side.

$$x+2<2x+14$$

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

$$x-2x<-2+14$$

Combine like terms.

$$-x<12$$

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

$$x>-12$$

Consider condition x>-7 specified above.

$$x>-7$$

Now consider the case when x+7 is negative. Move 7 to the right hand side.

$$x<-7$$

The initial inequality changes the direction when multiplied by x+7 for x+7<0.

$$x+2>2\left(x+7\right)$$

Multiply out the right hand side.

$$x+2>2x+14$$

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

$$x-2x>-2+14$$

Combine like terms.

$$-x>12$$

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

$$x<-12$$

Consider condition x<-7 specified above. The result remains the same.

$$x<-12$$

The final solution is the union of the obtained solutions.

$$x\in \left(-\infty,-12\right)\cup \left(-7,\infty\right)$$