Solve x+3/x-5>(0,5)

$$\frac { x + 3 } { x - 5 } > ( 0,5 )$$

Solve for x

$$x\in \left(-\infty;-11\right)\cup \left(5;\infty\right)$$

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.

$$x+3>0,5\left(x-5\right)$$

Multiply out the right hand side.

$$x+3>0,5x-2,5$$

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

$$x-0,5x>-3-2,5$$

Combine like terms.

$$0,5x>-5,5$$

Divide both sides by 0,5. Since 0,5 is positive, the inequality direction remains the same.

$$x>-11$$

Consider condition x>5 specified above.

$$x>5$$

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.

$$x+3<0,5\left(x-5\right)$$

Multiply out the right hand side.

$$x+3<0,5x-2,5$$

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

$$x-0,5x<-3-2,5$$

Combine like terms.

$$0,5x<-5,5$$

Divide both sides by 0,5. Since 0,5 is positive, the inequality direction remains the same.

$$x<-11$$

Consider condition x<5 specified above. The result remains the same.

$$x<-11$$

The final solution is the union of the obtained solutions.

$$x\in \left(-\infty;-11\right)\cup \left(5;\infty\right)$$