Solve x-2/x-4leq3

$$\frac { x - 2 } { x - 4 } \leq 3$$

Solve for x

$$x\in (-\infty,4)\cup [5,\infty)$$

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.

$$x-2\leq 3\left(x-4\right)$$

Multiply out the right hand side.

$$x-2\leq 3x-12$$

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

$$x-3x\leq 2-12$$

Combine like terms.

$$-2x\leq -10$$

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

$$x\geq 5$$

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.

$$x-2\geq 3\left(x-4\right)$$

Multiply out the right hand side.

$$x-2\geq 3x-12$$

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

$$x-3x\geq 2-12$$

Combine like terms.

$$-2x\geq -10$$

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

$$x\leq 5$$

Consider condition x<4 specified above.

$$x<4$$

The final solution is the union of the obtained solutions.

$$x\in (-\infty,4)\cup [5,\infty)$$