Solve 2-1/x>1

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

Solve for x

$$x\in \left(-\infty,0\right)\cup \left(1,\infty\right)$$

Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{x}{x}.

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

Since \frac{2x}{x} and \frac{1}{x} have the same denominator, subtract them by subtracting their numerators.

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

Factor out x.

$$\frac{1}{x}\left(x-1\right)>0$$

For the product to be positive, x and x-1 have to be both negative or both positive. Consider the case when x and x-1 are both negative.

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

The solution satisfying both inequalities is x<0.

$$x<0$$

Consider the case when x and x-1 are both positive.

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

The solution satisfying both inequalities is x>1.

$$x>1$$

The final solution is the union of the obtained solutions.

$$x<0\text{; }x>1$$