Problem 59 - Intermediate Algebra

Question

Add or subtract, and then simplify, if possible. See Example 7. $$\frac{5 x}{x-3}+\frac{4 x}{3-x}$$

Step-by-Step Solution

Answer

\( \frac{x}{x-3} \)

Step 1: Recognize Equivalent Denominators

Observe that the two fractions, \( \frac{5x}{x-3} \) and \( \frac{4x}{3-x} \), have denominators that are negatives of each other. Notice that \( 3 - x = -(x - 3) \).

Step 2: Adjust the Second Term

Multiply the numerator and denominator of the second fraction by -1 to make the denominators the same. This gives us \( \frac{4x}{3-x} = \frac{-4x}{x-3} \).

Step 3: Combine the Fractions

Now that we have a common denominator, we can add the two fractions: \( \frac{5x}{x-3} + \frac{-4x}{x-3} = \frac{5x - 4x}{x-3} \).

Step 4: Simplify the Expression

Subtract the numerators: \( 5x - 4x = x \), so the expression simplifies to \( \frac{x}{x-3} \).

Key Concepts

Adding Fractions

Adding fractions with algebraic expressions may feel a bit daunting at first, but it's quite similar to adding numerical fractions. The main goal is to add or subtract the numerators, while maintaining a common denominator. Let's break this down:

• Identify the fractions you need to work with – in our example, they are: \( \frac{5x}{x-3} \) and \( \frac{4x}{3-x} \).
• Before you can add the fractions, you need to ensure they have the same denominator. This requires establishing a common ground, which we'll address in the next section.

Remember, if the denominators are significantly different, finding a common denominator is essential. Rewriting the fractions might be required to perform the addition smoothly.

Common Denominator

Finding a common denominator is a crucial step when it comes to adding fractions. It ensures that both fractions have a common base so their numerators can be directly combined.In cases like our exercise, the fractions \( \frac{5x}{x-3} \) and \( \frac{4x}{3-x} \) have denominators which are negative equivalents.
This means \( 3-x = -(x-3) \), and a simple multiplication of the second fraction’s numerator and denominator by -1 will align them both to \( x-3 \).

• This changes \( \frac{4x}{3-x} \) to \( \frac{-4x}{x-3} \).
• Once the fractions share the denominator \( x-3 \), they can be directly added or subtracted as needed.

Finding a common denominator is simplifying the fractions to a common base, hence allowing seamless addition of their numerators.

Simplifying Expressions

Once you've established a common denominator and added the numerators, simplification is the next critical step. It's all about making the expression as straightforward as possible.In the given exercise, after combining the fractions:

• The operation \( \frac{5x - 4x}{x-3} \) simplifies the numerators to \( x \).
• This leaves you with the simplified fraction \( \frac{x}{x-3} \).

This simplification helps you understand the essential nature of the expression without unnecessary complexity, making the mathematical results more intuitive and easy to work with. Always check if further simplification or reduction of terms is possible to achieve the most concise expression.