Solve x+3/x-2>1

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

Solve for x

$$x>2$$

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.

$$x+3>x-2$$

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

$$x-x>-3-2$$

Combine like terms.

$$0>-5$$

Consider condition x>2 specified above.

$$x>2$$

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.

$$x+3<x-2$$

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

$$x-x<-3-2$$

Combine like terms.

$$0<-5$$

Consider condition x<2 specified above.

$$x\in \emptyset$$

The final solution is the union of the obtained solutions.

$$x>2$$