Problem 43 - Elementary Algebra

Question

For the following problems, add or subtract the rational expressions. $$ \frac{8 x-1}{x+2}-\frac{15 x+7}{x+2} $$

Step-by-Step Solution

Answer

Answer: The simplified rational expression is $\frac{-7x - 8}{x + 2}$.

Step 1: Identify the common denominator

We first need to identify the common denominator. In this case, both fractions have the same denominator, \((x+2)\): $$ \frac{8 x-1}{x+2}-\frac{15 x+7}{x+2} $$

Step 2: Combine the numerators

Since the denominators are the same, we can directly subtract the numerators: $$ \frac{(8x - 1) - (15x + 7)}{x + 2} $$

Step 3: Simplify the numerator

Now, we'll simplify the numerator by combining like terms: $$ \frac{8x - 1 - 15x - 7}{x+2} $$ $$ \frac{-7x - 8}{x + 2} $$

Step 4: Write the final answer

Our rational expression, after subtraction and simplification, is: $$ \frac{-7x - 8}{x + 2} $$

Key Concepts

Common Denominator

When working with rational expressions, just like with regular fractions, the first thing you should check is the denominator. The "denominator" refers to the part of the fraction below the line. In order to perform arithmetic operations such as addition or subtraction, the fractions must share a common denominator.

Finding a common denominator simplifies the process, enabling you to manage the fractions as if they were parts of a whole number. In the provided exercise, both fractions already have the same denominator,

• \(x + 2\)

This means we can safely move forward to the next step, which involves subtracting the numerators.

Subtracting Fractions

Subtracting fractions is similar to subtracting whole numbers, but it requires attention to the numerator alone when the denominators match. Once you've established a common denominator, as we have here with \(x + 2\), focus shifts to the numerators.

The process involves a straightforward subtraction of the numerators. Take the first numerator and subtract the second numerator from it:

• First Numerator: \(8x - 1\)
• Second Numerator: \(15x + 7\)

These numerators are placed over the common denominator:

• \(\frac{(8x - 1) - (15x + 7)}{x + 2}\)

Make sure to distribute any negative signs properly through the terms of the second numerator during subtraction.

Simplifying Expressions

Simplifying expressions entails combining like terms in the numerator after the subtraction has taken place. This step enhances clarity and reduces the expression to its simplest form.

In our exercise, observe the subtraction and how terms are combined:

• Subtract: \(8x - 1 - 15x - 7\)
• Combine like terms: \(-7x - 8\)

Now, the expression has been simplified. The final format for the rational expression is expressed as:

• \(\frac{-7x - 8}{x + 2}\)

This expression cannot be simplified any further unless specified numbers for \(x\) lead to additional factorization. Always check for this possibility, especially with more complex denominators.