Problem 19 - Algebra for College Students

Question

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ \frac{x-1}{2}+\frac{x+3}{3} $$

Step-by-Step Solution

Answer

The simplified sum is \(\frac{5x+3}{6}\).

Step 1: Identify the Problem Type

The problem involves adding two rational expressions \(\frac{x-1}{2}\) and \(\frac{x+3}{3}\). To do this, we need a common denominator.

Step 2: Determine the Least Common Denominator (LCD)

The denominators are 2 and 3. To find the least common denominator, find the least common multiple of these numbers, which is 6.

Step 3: Rewrite Each Fraction with the LCD

Convert each fraction to have the LCD as the new denominator: \(\frac{x-1}{2} = \frac{3(x-1)}{6}\) and \(\frac{x+3}{3} = \frac{2(x+3)}{6}\).

Step 4: Simplify the Numerators

Multiply the numerators by the factors needed to achieve the LCD: \(3(x-1) = 3x - 3\) and \(2(x+3) = 2x + 6\).

Step 5: Add the Numerators

Add the resulting fractions: \(\frac{3x-3}{6} + \frac{2x+6}{6} = \frac{3x - 3 + 2x + 6}{6}\).

Step 6: Combine Like Terms in the Numerator

Combine like terms in the numerator: \((3x + 2x) - 3 + 6 = 5x + 3\). This yields the expression \(\frac{5x+3}{6}\).

Step 7: Simplify the Rational Expression

Check if further simplification is possible. In this case, \(\frac{5x+3}{6}\) is already in its simplest form.

Key Concepts

Least Common Denominator

When dealing with rational expressions, like the fractions in this problem, it's crucial to find the Least Common Denominator (LCD) to combine them. The least common denominator is essentially the smallest number that both denominators can divide into evenly.
Let's look at our problem: the denominators were 2 and 3. To find the LCD, you need the least common multiple of these numbers.

• List the multiples of 2: 2, 4, 6, 8,...
• List the multiples of 3: 3, 6, 9, 12,...

You'll notice that the smallest number both lists share is 6. Therefore, 6 is the least common denominator.
This concept is important because it allows you to rewrite fractions in a way that makes them easier to add or subtract.

Fraction Addition

Adding fractions, like the rational expressions in our problem, requires a shared denominator. Once you've determined the least common denominator, the next step is to rewrite each fraction using that common denominator.
Using our example, we took the rational expressions \( \frac{x-1}{2} \) and \( \frac{x+3}{3} \), and transformed them:

• Multiply the numerator and the denominator of \( \frac{x-1}{2} \) by 3 to get \( \frac{3(x-1)}{6} \)
• Multiply the numerator and the denominator of \( \frac{x+3}{3} \) by 2 to get \( \frac{2(x+3)}{6} \)

Now both fractions have the same denominator, which makes the addition straightforward. Simply add the numerators over the common denominator:
\[ \frac{3x-3}{6} + \frac{2x+6}{6} = \frac{3x - 3 + 2x + 6}{6} \]
This is how fractions are added once they share a denominator.

Simplifying Fractions

After obtaining the combined expression, it's crucial to simplify it into its simplest form. Simplifying helps to see the expression clearly and makes further computation easier.
In our problem, after adding the numerators, we have the expression \( \frac{5x+3}{6} \).

• Combining like terms is an essential part of simplifying. Here, the terms \(3x\) and \(2x\) are like terms and combine to form \(5x\).
• Likewise, the constants \(-3\) and \(+6\) simplify to \(+3\).

This gives the final expression \( \frac{5x+3}{6} \), which is already in simplest form since there are no common factors in the numerator and the denominator that can be further reduced.
The key takeaway is to always check if the resulting fraction can be simplified further after adding or subtracting.