Solve x+21/x-4<1

$$\frac { x + 21 } { x - 4 } < 1$$

Solve for x

$$x<4$$

Solution Steps

Denominator x-4 cannot be zero since division by zero is not defined. There are two cases.

$$x-4>0 x-4<0$$

Consider the case when x-4 is positive. Move -4 to the right hand side.

$$x>4$$

The initial inequality does not change the direction when multiplied by x-4 for x-4>0.

$$x+21<x-4$$

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

$$x-x<-21-4$$

Combine like terms.

$$0<-25$$

Consider condition x>4 specified above.

$$x\in \emptyset$$

Now consider the case when x-4 is negative. Move -4 to the right hand side.

$$x<4$$

The initial inequality changes the direction when multiplied by x-4 for x-4<0.

$$x+21>x-4$$

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

$$x-x>-21-4$$

Combine like terms.

$$0>-25$$

Consider condition x<4 specified above.

$$x<4$$

The final solution is the union of the obtained solutions.

$$x<4$$