Problem 8 - Algebra A Combined Function

Question

Perform each indicated operation. Simplify if possible. \(\frac{8}{x+4}-\frac{3}{3 x+12}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{7}{x+4}\).

Step 1: Find a Common Denominator

The denominators in the given expression are \(x+4\) and \(3x+12\). Notice that \(3x+12\) can be factored further. \(3x+12 = 3(x+4)\). Thus, the common denominator for these fractions is \(3(x+4)\).

Step 2: Rewrite Each Fraction Using the Common Denominator

Rewrite the fractions such that both have the common denominator \(3(x+4)\). The first fraction becomes \(\frac{8}{x+4} \cdot \frac{3}{3} = \frac{24}{3(x+4)}\). The second fraction is already \(\frac{3}{3(x+4)}\).

Step 3: Subtract the Numerators

Now that both fractions have the same denominator, subtract the numerators: \(\frac{24}{3(x+4)} - \frac{3}{3(x+4)} = \frac{24 - 3}{3(x+4)}\).

Step 4: Simplify the Result

Simplify the numerator: \(24 - 3 = 21\). Thus, the expression simplifies to: \(\frac{21}{3(x+4)}\).

Step 5: Simplify Further if Possible

Since \(21\) and \(3\) have a common factor of \(3\), further simplify \(\frac{21}{3(x+4)}\) to \(\frac{7}{x+4}\).

Key Concepts

Common Denominator

When working with algebraic fractions, finding a common denominator is crucial for operations like addition or subtraction. A common denominator is a shared multiple of the denominators within a set of fractions.
In the exercise, the original denominators are \(x+4\) and \(3x+12\). We notice that \(3x+12\) can be factored into \(3(x+4)\). By factoring, we see that the expression has a common component, \(x+4\).
This enables us to determine the common denominator for our expressions, which is \(3(x+4)\). Using a common denominator allows for straightforward manipulation of the fractions, making it possible to perform operations on them.

Factoring Expressions

Factoring is the process of breaking down an algebraic expression into simpler terms, or 'factors,' which when multiplied together give the original expression. For example, \(3x+12\) can be rewritten as \(3(x+4)\), because both terms in \(3x+12\) are divisible by 3.
This technique is particularly useful for simplifying expressions and solving equations. In this exercise, factoring \(3x+12\) reveals the shared element \(x+4\), which helps in finding the common denominator.
Remember, factoring expressions helps in streamlining operations and is a fundamental skill in algebra.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form, where no further division of numerator and denominator is possible. This makes the fraction easier to work with and understand.
In this problem, after subtracting the fractions, we ended up with \(\frac{21}{3(x+4)}\). To simplify, identify that 21 and 3 have a common factor of 3. Divide both the numerator and the denominator by this common factor to achieve \(\frac{7}{x+4}\).
Simplifying not only clarifies calculations but also reveals the most reduced form of your answer. It's a step not to overlook!

Subtracting Fractions

Subtracting fractions is similar to adding fractions in that you must have a common denominator to directly combine numerators. Once a common denominator is established, you simply subtract the numerators while keeping the denominator constant.
As seen in the exercise, after rewriting both fractions with the common denominator \(3(x+4)\), we subtracted the numerators: \(24 - 3\).
This resulted in a new fraction \(\frac{21}{3(x+4)}\), which could then be further simplified. Remember, having a solid grasp of finding and using common denominators is key to accurately subtracting fractions.