Problem 59 - Prealgebra and Introductory Algebra

Question

Simplify. $$\frac{2}{x-3}+\frac{5}{x-4}$$

Step-by-Step Solution

Answer

\(\frac{7x - 23}{(x - 3)(x - 4)}\)

Step 1: Determine the Common Denominator

We need to have the same denominator in both fractions before we can sum them. In this case, the common denominator would be the product of the two denominators, which is \((x - 3)(x - 4)\).

Step 2: Express each Fraction with the Common Denominator and Consolidate

To express each fraction with the common denominator, multiply the numerator and denominator of the first fraction by \((x - 4)\), and the numerator and denominator of the second fraction by \((x - 3)\). This gives us \(\frac{2(x - 4)}{(x - 3)(x - 4)} + \frac{5(x - 3)}{(x - 3)(x - 4)}\). Now that both fractions have the same denominator, we can add the numerators: \(\frac{2(x - 4) + 5(x - 3)}{(x - 3)(x - 4)}\)

Step 3: Simplify the Expression

Now simplify the expression: \(\frac{2x - 8 + 5x - 15}{(x - 3)(x - 4)}\). This simplifies to \(\frac{7x - 23}{(x - 3)(x - 4)}\)

Key Concepts

Common Denominator

When adding algebraic fractions, a common denominator is essential. This allows the fractions to "line up," making addition possible. Think of it as finding a common ground for dialogue, where all parts speak the same "language." The denominators in each fraction represent this language. To start, identify each fraction's denominator. In our problem, the denominators are \(x-3\) and \(x-4\). The most effective method for finding a common denominator is to multiply these denominators together:

• Multiply \((x-3)(x-4)\).
• This product, \((x-3)(x-4)\), is the common denominator used for both fractions.

This process transforms the fractions, enabling addition. Carefully apply this technique to any algebraic fraction that you encounter to make addition possible.

Fraction Addition

Adding fractions with a common denominator simplifies the process of combining them. First, adjust each fraction so they share the same denominator, which we established earlier. Here's how:

• For \(\frac{2}{x-3}\), multiply the numerator and denominator by \(x-4\).
• For \(\frac{5}{x-4}\), multiply the numerator and denominator by \(x-3\).

This gives us:

• \(\frac{2(x-4)}{(x-3)(x-4)}\)
• \(\frac{5(x-3)}{(x-3)(x-4)}\)

Now that both fractions have the same denominator, you can add the numerators to unify the expression. The result is a single fraction: \(\frac{2(x-4) + 5(x-3)}{(x-3)(x-4)}\). Remember, fraction addition is much simpler when there are no differing denominators to worry about.

Expression Simplification

Once the fractions have been combined, the resulting expression can be simplified. Our expression after adding the fractions is \(\frac{2(x-4) + 5(x-3)}{(x-3)(x-4)}\).Begin simplifying the numerator:

• Distribute constants across the terms:
  • \(2(x-4) = 2x - 8\)
  • \(5(x-3) = 5x - 15\)
• Combine the results: \(2x - 8 + 5x - 15\)
• Simplify further by combining like terms: \(7x - 23\)

So, the entire expression becomes \(\frac{7x - 23}{(x-3)(x-4)}\). This process of expression simplification helps make calculations more manageable and cleans up the final result.