Solve x/x-2>5

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

Solve for x

$$x\in \left(2,\frac{5}{2}\right)$$

Solution Steps

Denominator x-2 cannot be zero since division by zero is not defined. There are two cases.

$$x-2>0 x-2<0$$

Consider the case when x-2 is positive. Move -2 to the right hand side.

$$x>2$$

The initial inequality does not change the direction when multiplied by x-2 for x-2>0.

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

Multiply out the right hand side.

$$x>5x-10$$

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

$$x-5x>-10$$

Combine like terms.

$$-4x>-10$$

Divide both sides by -4. Since -4 is negative, the inequality direction is changed.

$$x<\frac{5}{2}$$

Consider condition x>2 specified above.

$$x\in \left(2,\frac{5}{2}\right)$$

Now consider the case when x-2 is negative. Move -2 to the right hand side.

$$x<2$$

The initial inequality changes the direction when multiplied by x-2 for x-2<0.

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

Multiply out the right hand side.

$$x<5x-10$$

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

$$x-5x<-10$$

Combine like terms.

$$-4x<-10$$

Divide both sides by -4. Since -4 is negative, the inequality direction is changed.

$$x>\frac{5}{2}$$

Consider condition x<2 specified above.

$$x\in \emptyset$$

The final solution is the union of the obtained solutions.

$$x\in \left(2,\frac{5}{2}\right)$$