Solve +x+1/x-2>3

$$\frac { + x + 1 } { x - 2 } > 3$$

Solve for x

$$x\in \left(2,\frac{7}{2}\right)$$

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.

$$x+1>3\left(x-2\right)$$

Multiply out the right hand side.

$$x+1>3x-6$$

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

$$x-3x>-1-6$$

Combine like terms.

$$-2x>-7$$

Divide both sides by -2. Since -2 is negative, the inequality direction is changed.

$$x<\frac{7}{2}$$

Consider condition x>2 specified above.

$$x\in \left(2,\frac{7}{2}\right)$$

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.

$$x+1<3\left(x-2\right)$$

Multiply out the right hand side.

$$x+1<3x-6$$

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

$$x-3x<-1-6$$

Combine like terms.

$$-2x<-7$$

Divide both sides by -2. Since -2 is negative, the inequality direction is changed.

$$x>\frac{7}{2}$$

Consider condition x<2 specified above.

$$x\in \emptyset$$

The final solution is the union of the obtained solutions.

$$x\in \left(2,\frac{7}{2}\right)$$