Solve 3x-8/x-2leq1

$$\frac { 3 x - 8 } { x - 2 } \leq 1$$

Solve for x

$$x\in (2,3]$$

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.

$$3x-8\leq x-2$$

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

$$3x-x\leq 8-2$$

Combine like terms.

$$2x\leq 6$$

Divide both sides by 2. Since 2 is positive, the inequality direction remains the same.

$$x\leq 3$$

Consider condition x>2 specified above.

$$x\in (2,3]$$

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.

$$3x-8\geq x-2$$

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

$$3x-x\geq 8-2$$

Combine like terms.

$$2x\geq 6$$

Divide both sides by 2. Since 2 is positive, the inequality direction remains the same.

$$x\geq 3$$

Consider condition x<2 specified above.

$$x\in \emptyset$$

The final solution is the union of the obtained solutions.

$$x\in (2,3]$$