Problem 55 - Introductory Algebra for College Students

Question

Perform the indicated operations. If possible, simplify the answer. \(\frac{2 x+3}{x^{2}-7 x+12}-\frac{2}{x-3}(\) Section 7.4 , Example 7 )

Step-by-Step Solution

Answer

The simplified expression is: \(\frac{11}{(x-4)(x-3)}\)

Step 1: Identify Common Factors

First, factorize \( x^2 - 7x + 12 \) in the denominator of the first fraction. It can be written as \( (x-4)(x-3) \). Now, the expression becomes \( \frac{2x + 3}{(x-4)(x-3)} - \frac{2}{x-3} \)

Step 2: Combine the Fractions

To combine the fractions, we need to have the same denominator. Multiply the second expression by \( x-4 \) at numerator and denominator. It becomes \( \frac{2x + 3}{(x-4)(x-3)} - \frac{2(x-4)}{(x-4)(x-3)} \)

Step 3: Combine Similar Terms

Now both fractions have the same denominator, subtract the numerators, have \( \frac{2x + 3 - 2(x-4)}{(x-4)(x-3)} \), which can be written \( \frac{2x + 3 - 2x +8}{(x-4)(x-3)} \).

Step 4: Simplify the Expression

Combine similar terms in the numerator, it can be simplified as \( \frac{11}{(x-4)(x-3)} \)

Key Concepts

Factoring Quadratics

Factoring quadratics is a method used to simplify expressions, especially when dealing with rational expressions. In this exercise, the quadratic expression \( x^2 - 7x + 12 \) needs to be factored to simplify the rational expression. Factoring involves writing the quadratic as a product of two binomials.

Let's break it down:

• Identify the coefficient of \( x^2 \), the linear term, and the constant term.
• For \( x^2 - 7x + 12 \), we need two numbers that multiply to 12 (the constant term) and add to -7 (the coefficient of \( x \)).
• The numbers -3 and -4 meet these criteria since \((-3) \times (-4) = 12\) and \((-3) + (-4) = -7\).
• This allows us to rewrite the expression as \((x - 4)(x - 3)\).

Factoring transforms a quadratic into a more manageable form, which reveals common denominators and simplifies the computation of rational expressions.

Common Denominator

When working with multiple fractions, having a common denominator facilitates operations such as addition and subtraction. In this exercise, we initially have two fractions with different denominators. The first fraction's denominator is \((x-4)(x-3)\), while the second one is \(x-3\).

To combine these fractions, we need to make their denominators identical. Here's how to do it:

• Identify the least common denominator (LCD) for the fractions. Here, the LCD is \((x-4)(x-3)\).
• Adjust the second fraction by multiplying both its numerator and denominator by \(x-4\) to match the first fraction.
• The second fraction becomes \(\frac{2(x-4)}{(x-4)(x-3)}\).

Ensuring a common denominator allows the fractions to be combined into a single expression, simplifying further arithmetic operations.

Simplifying Expressions

Simplifying expressions is a crucial step in rational expressions to achieve the most concise form. After obtaining a common denominator, you can focus on simplifying the expression.

Here's how to simplify in the given problem:

• With the common denominator, \((x-4)(x-3)\), subtract the numerators of the two fractions.
• The numerators \(2x + 3\) and \(2(x-4)\) are simplified to \(2x + 3 - 2(x - 4)\).
• This simplifies to \(2x + 3 - 2x + 8\), which further simplifies to \(11\).
• Hence, the expression becomes \(\frac{11}{(x-4)(x-3)}\).

By simplifying expressions, we make them easier to understand and work with, leading to a clear and simplified answer.