Problem 57 - Introductory Algebra for College Students

Question

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{x-2}{x^{2}-25}-\frac{x-2}{25-x^{2}}$$

Step-by-Step Solution

Answer

The solution to the exercise is 0.

Step 1: Identify the Opposites

The denominators \(x^{2}-25\) and \(25-x^{2}\) are opposites of each other. This can be established by multiplying the second denominator by -1, resulting in the first denominator.

Step 2: Rewrite the Expression

Rewrite the expression as: \( \frac{x-2}{x^{2}-25}+ \frac{(x-2)(-1)}{x^{2}-25} \). Multiplying denominator by -1, converts subtraction into addition.

Step 3: Perform the Addition

The fractions both have the same denominator now, which means they can be added. Doing so gives: \( \frac{(x-2) - (x-2)}{x^{2}-25} \). As the numerator results in zero.

Step 4: Simplify the Expression

Simplify the expression to get the final answer: \( \frac{0}{x^{2} - 25} \), which simplifies further to 0 as any number (except undefined) divided by zero equals zero.

Key Concepts

Understanding Additive Inverses

The concept of additive inverses is a fundamental one in algebra. In simple terms, two numbers are additive inverses if their sum is zero. Imagine you have a number 'a,' its additive inverse would be '-a' because when you add 'a + (-a),' the result is zero.

Take, for example, the denominators in our exercise. The expressions x^2 - 25 and 25 - x^2 might look different at first glance, but they are indeed additive inverses. By multiplying one of them by -1, it transforms into the other. This switch is a clever manipulation that allows for the simplification of a complex algebraic fraction. Recognizing additive inverses quickly can save you from lengthy calculations and lead to quicker simplifications.

Finding Common Denominators

When working with fractions, whether simple or algebraic, finding common denominators is a crucial step in the process of adding or subtracting them. This concept is similar to finding common ground in an argument; it’s necessary for progress. A common denominator refers to a shared multiple of the denominators.

In the context of our exercise, we manipulated one of the fractions to turn the subtraction into an addition, thereby creating a common denominator. This is possible because algebraic fractions follow the same rules as numerical fractions. By securing a common denominator, we could directly compare and combine the numerators, ultimately simplifying the fraction's value. Remember, the goal is to combine like terms, which is only possible when the denominators match.

Simplifying Expressions

The art of simplifying expressions in algebra is all about making complex information more manageable and easier to understand. It's similar to decluttering a room—removing unnecessary items to make the space cleaner and more functional.

Here, after finding a common denominator, we simplify the expression. It often involves combining like terms, reducing fractions, and eliminating any mathematical clutter. In our example, simplifying led us to a dramatic conclusion: the numerator amounted to zero, meaning the entire expression equaled zero. This illustrates an important point: no matter how daunting an algebraic expression may seem, breaking it down step by step can lead to surprisingly simple outcomes. It's essential to remember that simplification may also involve factoring, expanding, or canceling terms to reach the most reduced form of an expression.