Solve x+4/x+12>10

$$\frac { x + 4 } { x + 12 } > 10$$

Solve for x

$$x\in \left(-\frac{116}{9},-12\right)$$

Solution Steps

Denominator x+12 cannot be zero since division by zero is not defined. There are two cases.

$$x+12>0 x+12<0$$

Consider the case when x+12 is positive. Move 12 to the right hand side.

$$x>-12$$

The initial inequality does not change the direction when multiplied by x+12 for x+12>0.

$$x+4>10\left(x+12\right)$$

Multiply out the right hand side.

$$x+4>10x+120$$

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

$$x-10x>-4+120$$

Combine like terms.

$$-9x>116$$

Divide both sides by -9. Since -9 is negative, the inequality direction is changed.

$$x<-\frac{116}{9}$$

Consider condition x>-12 specified above.

$$x\in \emptyset$$

Now consider the case when x+12 is negative. Move 12 to the right hand side.

$$x<-12$$

The initial inequality changes the direction when multiplied by x+12 for x+12<0.

$$x+4<10\left(x+12\right)$$

Multiply out the right hand side.

$$x+4<10x+120$$

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

$$x-10x<-4+120$$

Combine like terms.

$$-9x<116$$

Divide both sides by -9. Since -9 is negative, the inequality direction is changed.

$$x>-\frac{116}{9}$$

Consider condition x<-12 specified above.

$$x\in \left(-\frac{116}{9},-12\right)$$

The final solution is the union of the obtained solutions.

$$x\in \left(-\frac{116}{9},-12\right)$$