Problem 40 - Introductory Algebra for College Students

Question

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{6}{x-5}+\frac{2}{5-x}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(\frac{4}{x-5}\).

Step 1: Recognizing the Additive Inverses

The term \(x-5\) and \(5-x\) are additive inverses. We can rewrite \(5-x\) as \(-(x-5)\).

Step 2: Rewrite Equation with Common Denominator

We rewrite the fractions by treating \(x-5\) and \(-(x-5)\) as the same denominator. So, the expression evolves as follows: \(\frac{6}{x-5} - \frac{2}{x-5}\).

Step 3: Add the Fractions

Since they have the same denominator now, we can add the fractions: \(\frac{6-2}{x-5} = \frac{4}{x-5}\).

Key Concepts

Additive Inverses

The concept of additive inverses is foundational in understanding how to manipulate and simplify expressions, especially with variables. Additive inverses are pairs of numbers or expressions that sum up to zero. For example, in the expression from the exercise: \

• The terms \((x-5)\) and \((5-x)\) are additive inverses.
• When these expressions are added, their sum is zero.

This is due to the property that when you switch the order of numbers in subtraction, you effectively reverse their signs.

Notice that \(5-x\) can be rewritten as \\[\-(x-5)\\] \This transformation highlights the additive inverse nature, allowing us to manipulate and simplify the expression more easily. Remember, understanding additive inverses helps solve equations and simplify fractions by recognizing symmetrical terms that can be canceled out.

Common Denominator

Resolving expressions with fractions often requires rewriting them with a common denominator. A common denominator is a shared multiple of the denominators of two or more fractions, enabling them to be combined. In the given exercise, the terms \((x-5)\) and \\[-(x-5)\] \serve as a common denominator by recognizing them as identical in value once the inverse is considered.

When you rewrite the fractions as: \

• \((x-5)\) replaces both denominators, recognizing the equivalence.
• You express \(\frac{6}{x-5} - \frac{2}{x-5}\) with one denominator.

This shared denominator allows easy addition or subtraction of fractions, vital for simplifying the expression to a single fraction. Finding a common denominator involves identifying the least common multiple of the denominators, however, in this case, the additive inverse facilitated the process.

Simplifying Fractions

Simplifying fractions is an essential skill that involves reducing the fractions to their simplest form, where the numerator and denominator have no common factors other than 1. Once fractions have a common denominator, simplification is straightforward.

In the exercise, after establishing \((x-5)\) as the common denominator, the expression simplifies from:\[\\frac{6}{x-5} - \frac{2}{x-5} = \frac{6-2}{x-5} = \frac{4}{x-5}\\] Rather than dealing with two separate fractions, you are left with one that is easy to manage.

For complete simplification:

• Ensure the numerator is in its simplest form. Subtract similar terms when possible.
• Check for further potential reductions between new numerators and denominators.

Understanding how to simplify fractions helps make complex equations more manageable and leads to accurate results in algebra.