Solve 4x-4/x-2leq3

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

Solve for x

$$x\in [-2,2)$$

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.

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

Multiply out the right hand side.

$$4x-4\leq 3x-6$$

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

$$4x-3x\leq 4-6$$

Combine like terms.

$$x\leq -2$$

Consider condition x>2 specified above.

$$x\in \emptyset$$

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.

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

Multiply out the right hand side.

$$4x-4\geq 3x-6$$

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

$$4x-3x\geq 4-6$$

Combine like terms.

$$x\geq -2$$

Consider condition x<2 specified above.

$$x\in [-2,2)$$

The final solution is the union of the obtained solutions.

$$x\in [-2,2)$$