Problem 13 - Algebra for College Students

Question

Perform the indicated operations. Reduce answers to their lowest terms. See Example \(I\) $$ \frac{3 x-4}{2 x-4}+\frac{2 x-6}{2 x-4} $$

Step-by-Step Solution

Answer

\( \frac{5}{2} \)

Step 1: Combine the fractions

Since the denominators of both fractions are the same \(2x-4\), combine the numerators directly. The equation becomes: \[ \frac{(3x - 4) + (2x - 6)}{2x - 4} \]

Step 2: Simplify the numerator

Combine the like terms in the numerator: \((3x - 4) + (2x - 6) = 5x - 10\). The fraction is now: \[ \frac{5x - 10}{2x - 4} \]

Step 3: Factor the numerator and denominator

Factor out the greatest common factors in the numerator and denominator. \(5x - 10 = 5(x - 2)\) and \(2x - 4 = 2(x - 2)\). The fraction becomes: \[ \frac{5(x - 2)}{2(x - 2)} \]

Step 4: Cancel common factors

Cancel the common factor \(x - 2\) in both the numerator and the denominator: \[ \frac{5 \cancel{(x - 2)}}{2 \cancel{(x - 2)}} = \frac{5}{2} \]

Key Concepts

Combining Fractions

Combining fractions is simpler when the denominators are the same. In this case, you only need to add or subtract the numerators directly. This works because having a common denominator means each fraction is divided into the same-sized parts.
Let's look at the example provided: \[\frac{3x - 4}{2x - 4} + \frac{2x - 6}{2x - 4} \]
Since both fractions share the denominator \(2x - 4\), you can add the numerators together: \[ \frac{(3x - 4) + (2x - 6)}{2x - 4} \]
This results in: \[ \frac{5x - 10}{2x - 4} \]
Combining fractions with identical denominators reduces the problem to a simpler form, making the subsequent steps easier to handle.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves combining like terms. In the given problem, we started with two fractions and combined the numerators.
After combining, we had \( (3x - 4) + (2x - 6) = 5x - 10 \).
The next step was to simplify the expression by combining like terms, such as coefficients of \(x\) and constant terms. This simplification transforms complex fractions into more manageable equations. Always ensure the terms are correctly grouped to avoid mistakes in subsequent operations.

Factoring Polynomials

Factoring polynomials simplifies expressions by identifying common factors. Factoring transforms a polynomial into a product of its factors. In our fraction, the numerator \(5x - 10\) and the denominator \(2x - 4\) can be factored.
Factoring each: \[ 5x - 10 = 5(x - 2) \] and \[ 2x - 4 = 2(x - 2) \]
This reduces the initial fraction: \[ \frac{5(x - 2)}{2(x - 2)} \]
Factoring simplifies the process of cancelling common terms and makes the expression easier to manage.

Cancelling Common Factors

Cancelling common factors is a vital step in reducing fractions. It involves removing identical terms in the numerator and denominator.
In our problem, after factoring, we had \[ \frac{5(x - 2)}{2(x - 2)} \]
Both the numerator and the denominator have a common factor \(x - 2\), which can be cancelled out: \[ \frac{5 \cancel{(x - 2)} }{2 \cancel{(x - 2)} } = \frac{5}{2} \]
By cancelling \(x - 2\), we simplified the fraction to \( \frac{5}{2} \).
Cancelling common factors ensures the fraction is reduced to its simplest form, making it easier to understand and solve.