Solve 4x/1x-21-1geq3

$$\frac { 4 x } { 1 x - 21 - 1 } \geq 3$$

Solve for x

$$x\in (-\infty,-66]\cup (22,\infty)$$

Solution Steps

Subtract 1 from -21 to get -22.

$$\frac{4x}{1x-22}\geq 3$$

Denominator x-22 cannot be zero since division by zero is not defined. There are two cases.

$$x-22>0 x-22<0$$

Consider the case when x-22 is positive. Move -22 to the right hand side.

$$x>22$$

The initial inequality does not change the direction when multiplied by x-22 for x-22>0.

$$4x\geq 3\left(x-22\right)$$

Multiply out the right hand side.

$$4x\geq 3x-66$$

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

$$4x-3x\geq -66$$

Combine like terms.

$$x\geq -66$$

Consider condition x>22 specified above.

$$x>22$$

Now consider the case when x-22 is negative. Move -22 to the right hand side.

$$x<22$$

The initial inequality changes the direction when multiplied by x-22 for x-22<0.

$$4x\leq 3\left(x-22\right)$$

Multiply out the right hand side.

$$4x\leq 3x-66$$

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

$$4x-3x\leq -66$$

Combine like terms.

$$x\leq -66$$

The final solution is the union of the obtained solutions.

$$x\in (-\infty,-66]\cup (22,\infty)$$