Solve 5-3/x<1

$$5- \frac{ 3 }{ x } < 1$$

Solve for x

$$x\in \left(0,\frac{3}{4}\right)$$

Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 5 times \frac{x}{x}.

$$\frac{5x}{x}-\frac{3}{x}<1$$

Since \frac{5x}{x} and \frac{3}{x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5x-3}{x}<1$$

Factor out x.

$$\frac{1}{x}\left(4x-3\right)<0$$

For the product to be negative, x and x-\frac{3}{4} have to be of the opposite signs. Consider the case when x is positive and x-\frac{3}{4} is negative.

$$x>0 x-\frac{3}{4}<0$$

The solution satisfying both inequalities is x\in \left(0,\frac{3}{4}\right).

$$x\in \left(0,\frac{3}{4}\right)$$

Consider the case when x-\frac{3}{4} is positive and x is negative.

$$x-\frac{3}{4}>0 x<0$$

This is false for any x.

$$x\in \emptyset$$

The final solution is the union of the obtained solutions.

$$x\in \left(0,\frac{3}{4}\right)$$