Solve (x+5/x-7)<1

$$( \frac { x + 5 } { x - 7 } ) < 1$$

Solve for x

$$x<7$$

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.

$$x+5<x-7$$

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

$$x-x<-5-7$$

Combine like terms.

$$0<-12$$

Consider condition x>7 specified above.

$$x\in \emptyset$$

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.

$$x+5>x-7$$

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

$$x-x>-5-7$$

Combine like terms.

$$0>-12$$

Consider condition x<7 specified above.

$$x<7$$

The final solution is the union of the obtained solutions.

$$x<7$$