Solve x-7/x+5>2

$$\frac{ x-7 }{ x+5 } > 2$$

Solve for x

$$x\in \left(-17,-5\right)$$

Solution Steps

Denominator x+5 cannot be zero since division by zero is not defined. There are two cases.

$$x+5>0 x+5<0$$

Consider the case when x+5 is positive. Move 5 to the right hand side.

$$x>-5$$

The initial inequality does not change the direction when multiplied by x+5 for x+5>0.

$$x-7>2\left(x+5\right)$$

Multiply out the right hand side.

$$x-7>2x+10$$

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

$$x-2x>7+10$$

Combine like terms.

$$-x>17$$

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

$$x<-17$$

Consider condition x>-5 specified above.

$$x\in \emptyset$$

Now consider the case when x+5 is negative. Move 5 to the right hand side.

$$x<-5$$

The initial inequality changes the direction when multiplied by x+5 for x+5<0.

$$x-7<2\left(x+5\right)$$

Multiply out the right hand side.

$$x-7<2x+10$$

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

$$x-2x<7+10$$

Combine like terms.

$$-x<17$$

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

$$x>-17$$

Consider condition x<-5 specified above.

$$x\in \left(-17,-5\right)$$

The final solution is the union of the obtained solutions.

$$x\in \left(-17,-5\right)$$