Solve 5-3/a<1/a

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

Solve for a

$$a\in \left(0,\frac{4}{5}\right)$$

Solution Steps

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

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

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

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

Subtract \frac{1}{a} from both sides.

$$\frac{5a-3}{a}-\frac{1}{a}<0$$

Since \frac{5a-3}{a} and \frac{1}{a} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5a-3-1}{a}<0$$

Combine like terms in 5a-3-1.

$$\frac{5a-4}{a}<0$$

For the quotient to be negative, 5a-4 and a have to be of the opposite signs. Consider the case when 5a-4 is positive and a is negative.

$$5a-4>0 a<0$$

This is false for any a.

$$a\in \emptyset$$

Consider the case when a is positive and 5a-4 is negative.

$$a>0 5a-4<0$$

The solution satisfying both inequalities is a\in \left(0,\frac{4}{5}\right).

$$a\in \left(0,\frac{4}{5}\right)$$

The final solution is the union of the obtained solutions.

$$a\in \left(0,\frac{4}{5}\right)$$