Problem 53 - Algebra for College Students

Question

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ \frac{-3}{4 n+5}-\frac{8}{3 n+5} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{-41n - 55}{(4n+5)(3n+5)} \).

Step 1: Identify the denominators

In the expression \( \frac{-3}{4n+5} - \frac{8}{3n+5} \), we have two denominators: \( 4n+5 \) and \( 3n+5 \). To subtract these rational expressions, we need a common denominator.

Step 2: Find the least common denominator (LCD)

The least common denominator of \( 4n+5 \) and \( 3n+5 \) is their product \((4n+5)(3n+5)\) because they do not have common factors.

Step 3: Adjust each fraction to have the LCD

Rewrite each fraction with the common denominator \((4n+5)(3n+5)\):- Multiply the numerator and denominator of the first fraction by \((3n+5)\) to get \( \frac{-3(3n+5)}{(4n+5)(3n+5)} \).- Multiply the numerator and denominator of the second fraction by \((4n+5)\) to get \( \frac{8(4n+5)}{(4n+5)(3n+5)} \).

Step 4: Rewrite the expression with a common denominator

Now the expression becomes:\[ \frac{-3(3n+5)}{(4n+5)(3n+5)} - \frac{8(4n+5)}{(4n+5)(3n+5)} \].

Step 5: Subtract the numerators

Subtract the numerators:\[ -3(3n+5) - 8(4n+5) \].Distribute to get:\[ -9n-15 - 32n-40 \].Combine like terms:\[ -9n - 32n - 15 - 40 = -41n - 55 \].

Step 6: Write the final expression

The final expression is:\[ \frac{-41n-55}{(4n+5)(3n+5)} \].Since there are no common factors between \(-41n-55\) and \((4n+5)(3n+5)\), the expression is in its simplest form.

Key Concepts

Understanding the Least Common Denominator

Rational expressions often require us to work with different denominators. To effectively add or subtract them, we must find a common denominator. This is called the Least Common Denominator (LCD). It is the smallest expression that can be evenly divided by each of the denominators involved in the expressions we are dealing with. In our example, the denominators are \(4n+5\) and \(3n+5\). Because they share no common factors, the LCD is simply their product: \((4n+5)(3n+5)\). This ensures that both fractions can be expressed in terms of the same denominator, which allows us to subtract them seamlessly. To find an LCD, follow these tips:

• List the factors of each denominator.
• Identify any common factors.
• Multiply the unique factors together.

Finding the LCD may seem complicated at first, but with practice, it becomes a straightforward process that simplifies dealing with rational expressions.

Mastering Subtracting Fractions

Once you have a common denominator, subtracting fractions is similar to working with the numerical fractions you learned early on in math. With rational expressions, the only difference is the presence of variables. In our exercise, both fractions \( \frac{-3(3n+5)}{(4n+5)(3n+5)} \) and \( \frac{8(4n+5)}{(4n+5)(3n+5)} \) have the same denominator, simplifying the subtraction process. Here's a step-by-step method:

• Keep the denominator the same.
• Subtract the numerators by distributing first: \(-3(3n+5) - 8(4n+5)\).
• Distribute inside the parentheses to simplify each term.
• Combine like terms to form a single expression.

Remember, the key is to focus on the numerators once the denominators are unified. With practice, subtracting fractions in rational expressions becomes intuitive.

Simplifying Expressions Fully

Simplification is the final, and quite satisfying, step when working with rational expressions. In this process, the goal is to reduce the expression to its simplest form. This involves further combining like terms or factoring. For our problem, the subtraction resulted in the numerator \(-41n-55\), and the denominator remained \((4n+5)(3n+5)\). Here’s how you can approach simplification:

• Look for common factors in the numerator and denominator.
• If there are any common factors, divide them out to simplify.
• If there are no common factors, as in our case, the expression is already simplified.

Simplification is vital as it makes expressions more manageable and prepares them for further mathematical operations. Always double-check for missed factoring opportunities, as this can sometimes reveal further simplifications.