Problem 82 - Elementary Algebra

Question

Why are \(\frac{3}{x-2}\) and \(\frac{3}{2-x}\) opposites? What should be the result of adding \(\frac{3}{x-2}\) and \(\frac{3}{2-x}\) ?

Step-by-Step Solution

Answer

The expressions are opposites and their sum is 0.

Step 1: Identify the Expressions

We are given two expressions: \(\frac{3}{x-2}\) and \(\frac{3}{2-x}\). Our task is to determine if they are opposites and to find their sum.

Step 2: Rearrange the Denominator

Observe that \(2 - x = -(x - 2)\). By substituting this into \(\frac{3}{2-x}\), we get: \[\frac{3}{2-x} = \frac{3}{-(x-2)} = -\frac{3}{x-2}\]. This shows that \(\frac{3}{x-2}\) and \(\frac{3}{2-x}\) are opposites.

Step 3: Add the Expressions

Since \(\frac{3}{2-x}\) is \(-\frac{3}{x-2}\), adding these two expressions means we have: \[\frac{3}{x-2} + \left(-\frac{3}{x-2}\right) = \frac{3}{x-2} - \frac{3}{x-2} = 0.\]

Step 4: Conclusion

The expressions \(\frac{3}{x-2}\) and \(\frac{3}{2-x}\) are indeed opposites as shown by their equivalent form and combining them results in 0.

Key Concepts

Opposite Expressions

In mathematics, opposite expressions occur when two expressions are equal in magnitude but have opposite signs. When added together, these expressions cancel each other out, resulting in zero. Let's consider the expressions \( \frac{3}{x-2} \) and \( \frac{3}{2-x} \).

To determine if they are opposites, we need to look closely at the denominators. Notice that \( 2-x \) can be rewritten as \(-(x-2)\). This subtle change highlights that \( \frac{3}{2-x} \) is essentially \(-\frac{3}{x-2} \).

Because they have the same magnitude but opposite signs, we confirm that these expressions are indeed opposites. Recognizing and identifying opposite expressions allows for efficient simplification, especially when adding or subtracting them.

Simplifying Fractions

Simplifying fractions is a crucial skill in algebra and beyond. It involves reducing fractions to their simplest form, where the numerator and denominator share no common factors other than 1. Let's see how this applies to our expressions.

For \( \frac{3}{x-2} \), the expression is already simplified, since 3 and \(x-2\) have no common factors. However, to understand the simplification fully, consider \( \frac{3}{2-x} \). By observing that \( 2-x \) is the opposite of \( x-2 \), we can transform the expression.

Thus, \( \frac{3}{2-x} = \frac{3}{-(x-2)} = -\frac{3}{x-2} \). This rewritten form is the simplest version when combined with \( \frac{3}{x-2} \) in an operation. Such skills in simplifying fractions or identifying common transformations are vital to succeed in algebra.

Adding Rational Expressions

Adding rational expressions involves combining expressions over a common denominator. Similar to adding numeric fractions, it's essential that these expressions share the same denominator, or have their denominators adjusted to ensure they do.

In this scenario, \( \frac{3}{x-2} \) and \( \frac{3}{2-x} \), or rather \(-\frac{3}{x-2} \), share the same modified denominator. This allows us to add them directly:

• \( \frac{3}{x-2} + \left(-\frac{3}{x-2}\right) \)

Since they are opposites, their sum is 0, as the terms cancel each other out perfectly.

One crucial takeaway is the importance of understanding the properties of rational expressions, especially when they appear to differ but are inherently linked by opposites. This ability to manipulate and simplify expressions is pivotal for more complex algebraic operations.