Solve 2x+1/x-1leq1

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

Solve for x

$$x\in [-2,1)$$

Solution Steps

Denominator x-1 cannot be zero since division by zero is not defined. There are two cases.

$$x-1>0 x-1<0$$

Consider the case when x-1 is positive. Move -1 to the right hand side.

$$x>1$$

The initial inequality does not change the direction when multiplied by x-1 for x-1>0.

$$2x+1\leq x-1$$

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

$$2x-x\leq -1-1$$

Combine like terms.

$$x\leq -2$$

Consider condition x>1 specified above.

$$x\in \emptyset$$

Now consider the case when x-1 is negative. Move -1 to the right hand side.

$$x<1$$

The initial inequality changes the direction when multiplied by x-1 for x-1<0.

$$2x+1\geq x-1$$

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

$$2x-x\geq -1-1$$

Combine like terms.

$$x\geq -2$$

Consider condition x<1 specified above.

$$x\in [-2,1)$$

The final solution is the union of the obtained solutions.

$$x\in [-2,1)$$