Solve -y+1/y-1geq0

$$- \frac { y + 1 } { y - 1 } \geq 0$$

Solve for y

$$y\in [-1,1)$$

Solution Steps

Divide both sides by -1. Since -1 is negative, the inequality direction is changed. Zero divided by any non-zero number gives zero.

$$\frac{y+1}{y-1}\leq 0$$

For the quotient to be ≤0, one of the values y+1 and y-1 has to be ≥0, the other has to be ≤0, and y-1 cannot be zero. Consider the case when y+1\geq 0 and y-1 is negative.

$$y+1\geq 0 y-1<0$$

The solution satisfying both inequalities is y\in \left[-1,1\right).

$$y\in [-1,1)$$

Consider the case when y+1\leq 0 and y-1 is positive.

$$y+1\leq 0 y-1>0$$

This is false for any y.

$$y\in \emptyset$$

The final solution is the union of the obtained solutions.

$$y\in [-1,1)$$