Solve x-1/x-frac{1{2}}leq0

$$\frac { x - 1 } { x - \frac { 1 } { 2 } } \leq 0$$

Solve for x

$$x\in (\frac{1}{2},1]$$

Solution Steps

For the quotient to be ≤0, one of the values x-1 and x-\frac{1}{2} has to be ≥0, the other has to be ≤0, and x-\frac{1}{2} cannot be zero. Consider the case when x-1\geq 0 and x-\frac{1}{2} is negative.

$$x-1\geq 0 x-\frac{1}{2}<0$$

This is false for any x.

$$x\in \emptyset$$

Consider the case when x-1\leq 0 and x-\frac{1}{2} is positive.

$$x-1\leq 0 x-\frac{1}{2}>0$$

The solution satisfying both inequalities is x\in \left(\frac{1}{2},1\right].

$$x\in (\frac{1}{2},1]$$

The final solution is the union of the obtained solutions.

$$x\in (\frac{1}{2},1]$$