Problem 64 - College Algebra Essentials

Question

Simplify each complex rational expression. $$\frac{1-\frac{1}{x}}{x y}$$

Step-by-Step Solution

Answer

The simplified form of the complex rational expression \(\frac{1-\frac{1}{x}}{xy}\) is \(\frac{1 - \frac{1}{x}}{xy}\).

Step 1: Simplify the Numerator

First, simplify the numerator which is a fraction. To do this, find a common denominator. Here, it's 'x'. Now, rewrite the numerator as a single fraction by subtracting like terms: \(1-\frac{1}{x}\) can be written as \(\frac{x}{x} - \frac{1}{x}\), which simplifies to \(\frac{x-1}{x}\).

Step 2: Rewrite the Complex Fraction

Next, rewrite the whole complex fraction as \(\frac{\frac{x-1}{x}}{xy}\) which simplifies to \(\frac{x-1}{x^2y}\). This form is easier to manipulate.

Step 3: Simplify the Fraction

Finally, simplify the whole fraction by dividing each term in the numerator by 'x': \(\frac{x-1}{x^2y} = \frac{1 - \frac{1}{x}}{xy}.\)