\( \frac { 1 + \frac { 1 } { x } } { \frac { 1 } { x } - x } , x \neq 0 , \)

Did you know that simplifying rational expressions like this one dates back to ancient civilizations? The Egyptians, for instance, utilized fractions with a keen understanding of geometry, often employing unit fractions, which are fractions with a numerator of one. Their methods laid foundational principles for algebra and can still be relevant in teaching today! In real-world applications, simplifying expressions can help in fields like engineering and economics where ratios are frequent. For example, when dealing with load ratios in structures or budgeting, being able to simplify and manipulate ratios quickly can lead to more efficient calculations and better decision-making. So, whether you're balancing a budget or designing a bridge, mastering those fractions is key!