Problem 38 - Elementary Algebra

Question

For the following problems, add or subtract the rational expressions. $$ \frac{a+2}{a-5}-\frac{a+3}{a-5} $$

Step-by-Step Solution

Answer

Question: Subtract the following rational expressions: $$\frac{a+2}{a-5} - \frac{a+3}{a-5}$$ Answer: $$\frac{-1}{a-5}$$

Step 1: Identify the common denominator

In this case, both rational expressions have the same denominator, which is \((a - 5)\). So the common denominator is \((a - 5)\).

Step 2: Combine the rational expressions

Next, we will combine the rational expressions by adding or subtracting their numerators while keeping the common denominator. In this case, we are subtracting the rational expressions: $$ \frac{a + 2}{a - 5} - \frac{a + 3}{a - 5} $$

Step 3: Subtract the numerators

Subtract the numerators of the two rational expressions, remembering to apply the subtraction to both terms in the second numerator: $$ \frac{(a + 2) - (a + 3)}{a - 5} $$

Step 4: Simplify the result

Distribute the negative sign to both terms in the second numerator and then combine like terms: $$ \frac{a + 2 - a - 3}{a - 5} $$ The \(a\) terms cancel out: $$ \frac{2 - 3}{a - 5} $$ Finally, we simplify and get: $$ \frac{-1}{a - 5} $$ So, the final result is: $$ \frac{-1}{a - 5} $$

Key Concepts

Adding Rational Expressions

Adding rational expressions involves finding a common denominator and then adding the numerators together, all while keeping that common denominator. A rational expression is a fraction where both the numerator and the denominator are polynomials. To add them:

• Identify the common denominator of the expressions. This is the least common multiple of the individual denominators.
• Adjust the numerators accordingly. If denominators are the same, simply add together the numerators.
• Combine the numerators, placing the result over the common denominator.
• Finally, simplify if necessary.

Remember, just like with numerical fractions, you cannot add rational expressions unless they have a common denominator. Keeping this in mind will make the process straightforward.

Subtracting Rational Expressions

Subtracting rational expressions is very similar to adding them but requires extra care in handling the negative sign during subtraction. Here's how you do it:

• Ensure that the denominators of the expressions are the same, which gives you a common denominator.
• Rewrite the expression so that both fractions share this common denominator.
• Subtract the numerators, keeping in mind to distribute the negative sign across the entire second numerator.
• Place the resulting numerator over the common denominator.
• Simplify the expression if possible.

In addition, always be careful with the negative sign, as it can change the entire result of the expression. Consistently check your work to avoid simple errors.

Common Denominator

The common denominator is crucial when it comes to adding and subtracting rational expressions. Think of it like a shared base that allows you to combine or distinguish between fractions. To find the common denominator:

• Look at the denominators of the expressions you are working with.
• If they are the same, as in the exercise, then that is your common denominator immediately.
• If they are different, find the least common multiple (LCM) of these denominators.
• Adjust the numerators as needed to reflect this shared denominator.

In many cases, the LCM can be found by multiplying the distinct factors of each denominator together, ensuring all essential factors are included to make the expressions compatible.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is the final key step in many algebra problems, including those involving rational expressions. By simplifying, you make the expression easier to understand and work with. Here's how to simplify:

• Combine like terms in the numerator, reducing it as much as possible.
• If possible, factor both the numerator and the denominator to see if there are common factors that can be canceled out.
• Ensure that your final expression is in its simplest form, particularly if any terms can be reduced.

In the original exercise, this involved canceling out terms that appeared in both the addition and subtraction steps. This reduced the expression to its simplest form, providing a clear and concise final answer.