Solve x-7/x+8>4

$$\frac { x - 7 } { x + 8 } > 4$$

Solve for x

$$x\in \left(-13,-8\right)$$

Solution Steps

Denominator x+8 cannot be zero since division by zero is not defined. There are two cases.

$$x+8>0 x+8<0$$

Consider the case when x+8 is positive. Move 8 to the right hand side.

$$x>-8$$

The initial inequality does not change the direction when multiplied by x+8 for x+8>0.

$$x-7>4\left(x+8\right)$$

Multiply out the right hand side.

$$x-7>4x+32$$

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

$$x-4x>7+32$$

Combine like terms.

$$-3x>39$$

Divide both sides by -3. Since -3 is negative, the inequality direction is changed.

$$x<-13$$

Consider condition x>-8 specified above.

$$x\in \emptyset$$

Now consider the case when x+8 is negative. Move 8 to the right hand side.

$$x<-8$$

The initial inequality changes the direction when multiplied by x+8 for x+8<0.

$$x-7<4\left(x+8\right)$$

Multiply out the right hand side.

$$x-7<4x+32$$

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

$$x-4x<7+32$$

Combine like terms.

$$-3x<39$$

Divide both sides by -3. Since -3 is negative, the inequality direction is changed.

$$x>-13$$

Consider condition x<-8 specified above.

$$x\in \left(-13,-8\right)$$

The final solution is the union of the obtained solutions.

$$x\in \left(-13,-8\right)$$