Problem 52 - Introductory Algebra for College Students

Question

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{11}{x+7}-\frac{5}{-x-7}$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{16}{x+7}\).

Step 1: Write the Given Expression

First, write the given expression, which is \(\frac{11}{x+7}-\frac{5}{-x-7}\). This is a subtraction of two rational expressions with opposite denominators.

Step 2: Use the property of opposite denominators

Use the property of opposite denominators, which states that \(\frac{a}{b} - \frac{c}{-b} = \frac{a+c}{b}\). So, apply this principle to the given expressions, and that gives \(\frac{11+5}{x+7}\).

Step 3: Simplify the Expression

Next, simplify the expression in the numerator. This simplifies to \(\frac{16}{x+7}\).

Key Concepts

Understanding Additive Inverses in Rational Expressions

In mathematics, additive inverses are numbers or expressions that, when added together, result in zero. This concept is crucial when simplifying rational expressions with opposite denominators. When you encounter opposite terms like \(x + 7\) and \(-x - 7\), they are additive inverses. The opposite signs indicate that they cancel each other out when combined properly.

Here's how it works:

• Identify each term or denominator in your expression.
• If you can rewrite one as the negative of the other, they are opposites.
• Using this property allows for simplifications that ease further calculations.

Recognizing additive inverses is a useful skill when working with rational expressions, as it helps simplify complex equations to their simplest form.

Dealing with Opposite Denominators

Opposite denominators occur when two denominators are additive inverses, such as \(x + 7\) and \(-x - 7\). Knowing how to handle these is essential for simplifying rational expressions effectively.

The rule you want to keep in mind is: \[\frac{a}{b} - \frac{c}{-b} = \frac{a + c}{b}\]This rule comes from being able to multiply both the numerator and the denominator of the second fraction by -1, transforming opposite denominators into the same denominator, thereby making subtraction or addition straightforward.

This approach allows for easier arithmetic processes and reduces the complexity of expressions.

Simplifying Rational Expressions

Once you're familiar with the rules regarding additive inverses and opposite denominators, simplifying rational expressions becomes a straightforward task. In our example, after using additive inverse properties, we combine the expressions:

• Add the numerators: \(11 + 5 = 16\)
• Keep the common denominator: \(x+7\)

Thus, our simplified expression is \(\frac{16}{x+7}\).

Simplifying expressions reduces their complexity, making them easier to evaluate or further manipulate. Understanding these processes allows solving algebraic expressions more efficiently and prevents errors in calculations. Being methodical in recognizing patterns and applying mathematical rules simplifies not only the current problem but also future ones.