Problem 52 - Elementary Algebra

Question

For the following problems, add or subtract the rational expressions. $$ \frac{x-1}{x-3}+\frac{x+4}{x-4} $$

Step-by-Step Solution

Answer

Question: Add the given rational expressions and simplify the result: $$ \frac{x-1}{x-3} + \frac{x+4}{x-4} $$ Answer: The sum of the given rational expressions is $$ \frac{2x^2 - 4x - 8}{(x-3)(x-4)} $$

Step 1: Identify the Least Common Denominator (LCD) of the rational expressions

The denominators in the given rational expressions are \((x-3)\) and \((x-4)\). Since there are no common factors, the LCD is the product of the two denominators: $$(x-3)(x-4)$$.

Step 2: Rewrite each fraction with the LCD as the new denominator

Multiply the first fraction by \(\frac{x-4}{x-4}\) and the second fraction by \(\frac{x-3}{x-3}\) to make both denominators equal to the LCD: $$ \frac{x-1}{x-3} \cdot \frac{x-4}{x-4} + \frac{x+4}{x-4} \cdot \frac{x-3}{x-3} = \frac{(x-1)(x-4)}{(x-3)(x-4)} + \frac{(x+4)(x-3)}{(x-4)(x-3)} $$

Step 3: Add the fractions with the same denominators

Since both fractions have the same denominator, you can add the numerators: $$ \frac{(x-1)(x-4) + (x+4)(x-3)}{(x-3)(x-4)} $$

Step 4: Expand and simplify the numerator

Distribute the terms in the numerators and combine like terms: $$ \frac{x^2 - 4x - x + 4 + x^2 - 3x + 4x - 12}{(x-3)(x-4)} = \frac{2x^2 - 4x + 4 - 12}{(x-3)(x-4)} $$ Now, simplify further by combining the constants: $$ \frac{2x^2 - 4x - 8}{(x-3)(x-4)} $$

Step 5: Check for any possible factorization and simplification

There is no further common factor between the numerator and denominator, so this is the final simplified answer: $$ \frac{2x^2 - 4x - 8}{(x-3)(x-4)} $$

Key Concepts

Least Common Denominator

When working with rational expressions, the first step in adding or subtracting them is finding a common ground for the denominators. This common ground is called the Least Common Denominator (LCD), and it is essential for bringing the expressions together. Think of it like finding a common language between two parties, allowing communication— in this case, addition or subtraction— to occur smoothly.

To determine the LCD, examine the denominators of the fractions involved. Sometimes, they might share common factors. However, if they don’t, as we saw in the exercise with \(x-3\) and \(x-4\), the LCD is simply the product of the two denominators. This means that we multiply them together: \( (x-3)(x-4) \). Always remember that the LCD might involve factoring, especially if the denominators were more complex.

Addition and Subtraction of Fractions

Once we have the Least Common Denominator, we need to rewrite each fraction so that they have this same denominator. This is similar to ensuring that all parts of a construction have the same foundation, which allows combining them seamlessly.

For the exercise provided, we multiplied each fraction by a specific form of 1, like \(\frac{x-4}{x-4}\) and \(\frac{x-3}{x-3}\). This method does not change the value of each fraction but alters its form so that they share the same denominator: \( (x-3)(x-4) \).

• After this step, the fractions look like this: \( \frac{(x-1)(x-4)}{(x-3)(x-4)} \) and \( \frac{(x+4)(x-3)}{(x-4)(x-3)} \).
• Once they have the same denominator, you can add or subtract them directly. In our case, this involves adding their numerators, creating a single fraction.

Simply perform the operation on the numerators while keeping the LCD as the common base, and there you have it— the combined expression.

Simplifying Expressions

Simplifying the resulting expression is the final and often the most important step. Now that we have a single fraction with the same denominator, it’s time to simplify it as much as possible. Simplification can make complex expressions much easier to work with and understand.

In this exercise, we expanded the numerators by distributing their terms across the binomials in the brackets, like so: \( x^2 - 4x - x + 4 \) and \( x^2 - 3x + 4x - 12 \).

• After expanding, combining like terms will lower the count of terms we need to consider, resulting in \( 2x^2 - 8 - 4x \).
• Check if the expression can be further factored or simplified. In our example, the numerator and denominator have no common factors, signaling completion of the simplification process.

This concise form, \ \frac{2x^2 - 4x - 8}{(x-3)(x-4)} \, is much easier to interpret and can help in understanding or applying these expressions in further mathematical problems.