Problem 42 - Introductory Algebra for College Students

Question

Divide as indicated. $$\frac{4}{x-6}+\frac{40}{7 x-42}$$

Step-by-Step Solution

Answer

The simplified expression after division is \(\frac{68}{7x-42}\).

Step 1: Identifying Common Denominators

First, observe the denominators in the given fraction. They are \(x-6\) and \(7x-42\). Note that \(7x-42\) is itself equal to \(7(x-6)\). This means our common denominator will be \(7(x-6)\) or equivalently \(7x-42\).

Step 2: Getting Common Denominators

Next, rewrite the given expression so that both fractions have the common denominator. This gives us:\[\frac{4}{x-6} \cdot \frac{7}{7}+\frac{40}{7x-42} = \frac{28}{7x-42}+\frac{40}{7x-42}\]

Step 3: Combining Fractions

Finally, add the fractions together, since they now have the same denominator:\[\frac{28}{7x-42}+\frac{40}{7x-42} = \frac{28+40}{7x-42} = \frac{68}{7x-42}\].

Key Concepts

Common Denominator

When working with algebraic fractions, one of the most important steps is finding a common denominator. A common denominator is a shared multiple of the denominators of two or more fractions. It allows us to rewrite each fraction so that they have comparable bases, making operations like addition and subtraction possible.

To find the common denominator, first consider the denominators in the given fractions. In this exercise, the denominators are \(x-6\) and \(7x-42\). Notice that \(7x-42\) can be factored into \(7(x-6)\). By identifying this common factor, \(x-6\) appears in both expressions
. The full common denominator becomes \(7(x-6)\) or \(7x-42\).

This process of finding a common denominator might involve factoring or identifying common factors as done here. Once you establish a common denominator, you're prepared to rewrite and manipulate fractions in subsequent steps.

Combining Fractions

Once fractions share a common denominator, you can easily combine them. This is because the denominators align, allowing the numerators to come together through addition or subtraction.

Let's decode this using the given exercise: the fractions \(\frac{4}{x-6}\) and \(\frac{40}{7x-42}\) are rewritten after establishing a common denominator, giving \(\frac{28}{7x-42}\) and \(\frac{40}{7x-42}\).

Combining means just adding the numerators over the shared denominator:

• Add the numerators: \(28 + 40\)
• Keep the common denominator: \(7x-42\)

After combining, the result is \(\frac{68}{7x-42}\). Remember, combining fractions is not possible unless they have the same denominator. This guarantees that we're working with equivalent portions.

Simplifying Expressions

Simplifying expressions is an essential step in dealing with fractions. It makes them easier to understand and compare. Simplification involves reducing an expression to its simplest form by factoring or canceling out any common factors.

In the expression \(\frac{68}{7x-42}\), we check if the numerator and denominator share any common factors that can be canceled out.

In simpler terms:

• Check the numbers in the numerator and denominator for GCD (Greatest Common Divisor).
• If possible, factor out common terms.

In this exercise, the expression doesn't simplify further because there are no additional common factors between the numerator 68 and the algebraically expressed denominator 7\(x-6\). Thus, it's safe to say that the expression \(\frac{68}{7x-42}\) is already in its simplest form.

Simplification ensures that the solution is as compact and efficient as possible, providing clarity and reducing computation complexity in future calculations.