Problem 54 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

Question: Add or subtract the given rational expressions, then simplify the result. $$ \frac{a+3}{a-3}-\frac{a+2}{a-2} $$ Answer: $$ \frac{2a}{(a-3)(a-2)} $$

Step 1: Find the common denominator

To find the common denominator, we will need to find the least common multiple (LCM) of the denominators \((a-3)\) and \((a-2)\). Since the two denominators are distinct linear factors, the LCM is simply the product of the two denominators: \((a-3)(a-2)\).

Step 2: Rewrite the fractions with the common denominator

With the common denominator of \((a-3)(a-2)\), we will rewrite both fractions with the new denominator by multiplying the numerator and the denominator by the missing factor: $$ \frac{a+3}{a-3}\cdot\frac{a-2}{a-2}-\frac{a+2}{a-2}\cdot\frac{a-3}{a-3}= \frac{(a+3)(a-2)}{(a-3)(a-2)}-\frac{(a+2)(a-3)}{(a-3)(a-2)} $$

Step 3: Combine the numerators

Now that both fractions have the same denominator, we can combine the numerators by carrying out the subtraction: $$ \frac{(a+3)(a-2)-(a+2)(a-3)}{(a-3)(a-2)} $$

Step 4: Expand the numerators

Before simplifying further, distribute the terms in the numerators: $$ \frac{(a^2-2a+3a-6)-(a^2-3a+2a-6)}{(a-3)(a-2)} $$ $$ \frac{a^2+a-6-(a^2-a-6)}{(a-3)(a-2)} $$

Step 5: Simplify the expression

Simplify by subtracting corresponding terms in the numerator, which will give us the final answer: $$ \frac{a^2+a-6-a^2+a+6}{(a-3)(a-2)}=\frac{2a}{(a-3)(a-2)} $$ The simplified expression is: $$ \frac{2a}{(a-3)(a-2)} $$

Key Concepts

Common Denominator

When you work with rational expressions, one key step in adding or subtracting them is finding a common denominator. A common denominator allows you to combine the fractions, just like when you work with regular fractions. Imagine having two pieces of cake that you want to add together. If one piece is cut into 3 equal parts, and the other into 4, it’s tricky to directly compare or add them. By finding a common way to cut all pieces (perhaps into 12 smaller parts each), you can easily see how they fit together.

For rational expressions like \( \frac{a+3}{a-3} \) and \( \frac{a+2}{a-2} \), this step involves looking at the denominators, which are \((a-3)\) and \((a-2)\). These denominators are distinct, meaning they're each a different factor. To find a shared base for combining them, you multiply them together to get \((a-3)(a-2)\), which serves as your common denominator.

Least Common Multiple

Finding the least common multiple (LCM) is a process that helps in simplifying the task of adding or subtracting rational expressions. The LCM is the smallest expression that both denominators can divide into without leaving a remainder. It serves as the perfect shared base for the denominators.

In the exercise, the denominators are \((a-3)\) and \((a-2)\). As these are linear polynomials with no common factors, the LCM is simply the product of these denominators, \( (a-3)(a-2) \). This LCM becomes the new denominator for both fractions, allowing them to be easily combined or compared.

Simplifying Expressions

Simplifying expressions is one of the last steps after rewriting fractions with a common denominator. It involves making the numerators easier to manage and reducing the expression to its simplest form.

• After rewriting the fractions with the common denominator \((a-3)(a-2)\), you'll need to combine their numerators.
• This often involves expanding the numerators and then carefully subtracting or adding them.

In our example, expand the numerators: - The first is \((a+3)(a-2)\), which distributes to \(a^2 - 2a + 3a - 6\).- The second is \((a+2)(a-3)\), distributing to \(a^2 - 3a + 2a - 6\).

Once expanded, subtract the second from the first: - Resulting in: \(a^2 + a - 6 - (a^2 - a - 6)\).- Simplifying this gives: \(2a\).This leads to the final simplified expression of: \( \frac{2a}{(a-3)(a-2)} \). This last fraction is in its simplest form, ready and neat!