Solve 2x-x/x-3leq1

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

Solve for x

$$x<3$$

Solution Steps

Combine 2x and -x to get x.

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

Denominator x-3 cannot be zero since division by zero is not defined. There are two cases.

$$x-3>0 x-3<0$$

Consider the case when x-3 is positive. Move -3 to the right hand side.

$$x>3$$

The initial inequality does not change the direction when multiplied by x-3 for x-3>0.

$$x\leq x-3$$

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

$$x-x\leq -3$$

Combine like terms.

$$0\leq -3$$

Consider condition x>3 specified above.

$$x\in \emptyset$$

Now consider the case when x-3 is negative. Move -3 to the right hand side.

$$x<3$$

The initial inequality changes the direction when multiplied by x-3 for x-3<0.

$$x\geq x-3$$

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

$$x-x\geq -3$$

Combine like terms.

$$0\geq -3$$

Consider condition x<3 specified above.

$$x<3$$

The final solution is the union of the obtained solutions.

$$x<3$$