Problem 37 - Elementary Algebra

Question

For the following problems, add or subtract the rational expressions. $$ \frac{b+1}{b-3}+\frac{b+2}{b-3} $$

Step-by-Step Solution

Answer

Question: Add the given fractions and simplify the result, if possible: $\frac{b+1}{b-3}+\frac{b+2}{b-3}$. Answer: The sum of the given rational expressions is $\frac{2b+3}{b-3}$.

Step 1: Identify the common denominator

The two fractions share a common denominator, which is \((b-3)\).

Step 2: Add the numerators

Add the numerators of the two fractions (\((b+1)\) and \((b+2)\)): \((b+1)+(b+2)\).

Step 3: Simplify the numerators

Combine the like terms in the sum of the numerators: \((b+1)+(b+2)=2b+3\)

Step 4: Write the sum as a single fraction

Place the simplified sum of the numerators \((2b+3)\) over the common denominator \((b-3)\): $$\frac{2b+3}{b-3}$$

Step 5: Check for further simplification

The resulting fraction can't be simplified further, so we have our final answer.

Step 6: Final Answer

The sum of the given rational expressions is: $$ \frac{b+1}{b-3}+\frac{b+2}{b-3} = \frac{2b+3}{b-3} $$

Key Concepts

Common Denominator

When working with rational expressions, finding a common denominator is crucial for combining fractions through addition or subtraction. The denominator is the bottom part of a fraction and must be the same in each fraction before performing these basic operations. In essence, the common denominator ensures you're working with parts of the whole that are identical, making the math straightforward and accurate.

In the given exercise:

• The fractions share a common denominator, \((b-3)\).
• This means that, as you're adding them, you don't need to change either fraction before proceeding.

By having this shared denominator from the start, you can focus on the numerators for the rest of the process. This simplifies the task significantly because no additional work is needed to align the fractions' denominators.

Simplifying Fractions

Once you add or multiply the fractions, simplifying them is the next step. Simplifying means reducing the fraction to its simplest form, making both the numerator and the denominator as small as possible while keeping the same value.

In this exercise, after adding the numerators:

• The sum was \((b+1) + (b+2) = 2b + 3\),
• forming a new fraction \(\frac{2b + 3}{b-3}\).

Upon checking, if there are no common factors between this numerator and the denominator, the fraction can't be simplified further. This means all numeric and variable parts have been reduced to their simplest form, offering a clear and concise final expression. Simplifying ensures your final answer is clean and easy to interpret.

Adding Fractions

Adding fractions with like denominators is a matter of simply combining the numerators since the fractional parts are already expressed in terms of the same whole. This operation goes as follows:

• Identify and confirm that the denominators match.
• Add the numerators together as separate from the denominator: \((b+1) + (b+2) = 2b + 3\).
• The sum remains over the common denominator \((b-3)\).

The key takeaway is the ease of adding fractions when the denominators are the same. It eliminates the need for extra steps to normalize the denominators, allowing you to directly perform addition. The calculated expression is clear, and final reduction checks are brief if the initial setup is correct. This means working efficiently and accurately with fractions becomes almost second nature once you master these concepts.