Problem 36 - Elementary Algebra

Question

For the following problems, add or subtract the rational expressions. $$ \frac{a-6}{a+2}+\frac{a-2}{a+2} $$

Step-by-Step Solution

Answer

Question: Simplify the following rational expression: \(\frac{a-6}{a+2}+\frac{a-2}{a+2}\). Answer: \(\frac{2a-8}{a+2}\)

Step 1: Identify the denominators

Here, the denominators for both fractions are \((a + 2)\). Since they are the same, we can proceed with adding the numerators.

Step 2: Add the numerators

Now, we add the numerators to obtain the sum of the rational expressions: $$ \frac{a-6}{a+2}+\frac{a-2}{a+2} = \frac{(a-6) + (a-2)}{a+2} $$

Step 3: Simplify the fraction

We simplify the expression by combining the like terms: $$ \frac{(a-6) + (a-2)}{a+2} = \frac{2a-8}{a+2} $$ The solution to the problem is: $$ \frac{a-6}{a+2}+\frac{a-2}{a+2} = \frac{2a-8}{a+2} $$

Key Concepts

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are both polynomials. In other words, they are divisions of polynomial expressions. A typical rational expression looks like \( \frac{P(x)}{Q(x)} \) where both \( P(x) \) and \( Q(x) \) are polynomial expressions. When adding rational expressions, it's important to understand that similar to numeric fractions, they can only be combined directly if they have the same denominator.

Working with rational expressions often involves finding a common denominator, and then simplifying the expression as you would with numerical fractions. For instance, in the example \( \frac{a-6}{a+2}+\frac{a-2}{a+2} \) the denominators are the same, enabling a straightforward addition of the numerators.

Common Denominators

The concept of common denominators is essential when adding or subtracting rational expressions. When the denominators of the expressions being added are different, a common denominator must be found. This is a value that both denominators can divide into without a remainder, essentially the least common multiple of the denominators.

In some cases, like in the exercise \( \frac{a-6}{a+2}+\frac{a-2}{a+2} \) you are already provided with a common denominator (\(a+2\) in both expressions), which simplifies the process. When the denominators are identical, the expressions can be combined by adding or subtracting the numerators, as they are over the same common division.

Simplify Fractions

After finding a common denominator and combining the numerators, the next step is to simplify the fraction. This is done by combining like terms in the numerator and then reducing the fraction to its simplest form. Simplification can involve factoring the numerator and denominator and cancelling any common factors that appear in both.

In our example, \( \frac{2a-8}{a+2} \) represents the expression post-combination of numerators. The simplification is straightforward since the expression cannot be reduced further; there are no common factors that can be cancelled between the numerator and the denominator. However, in other cases, simplifying might involve more steps such as factoring a polynomial to identify common factors.