Problem 39 - Prealgebra and Introductory Algebra

Question

Simplify. $$\frac{x-3}{6 x}+\frac{x+4}{8 x}$$

Step-by-Step Solution

Answer

\(\frac{7}{24}\)

Step 1: Identify denominators

In this case, the denominators are \(6x\) and \(8x\). We need to find the least common multiple of these two denominators to combine the fractions.

Step 2: Find the least common multiple

The least common multiple (LCM) of \(6x\) and \(8x\) is \(24x\).

Step 3: Rewrite fractions

To rewrite the original fractions with the new common denominator \(24x\), multiply the fraction \((x-3)/(6x)\) by \(4/4\), and multiply the fraction \((x+4)/(8x)\) by \(3/3\). This results into \(\frac{4x-12}{24x} + \frac{3x+12}{24x}\).

Step 4: Combine the fractions

Now that the fractions have the same denominator, they can be combined: \(\frac{4x-12+3x+12}{24x}\)..

Step 5: Simplify the numerator

By simplifying the above expression, the result is: \(\frac{7x}{24x}\).

Step 6: Simplify the fraction

The 'x' in the numerator and denominator can cancel out, leaving the simplified result of \(7/24\).

Key Concepts

Understanding the Least Common Multiple

The Least Common Multiple (LCM) is a fundamental concept when it comes to combining fractions, particularly when they have different denominators. In simple terms, the LCM of two or more numbers is the smallest number that all of them can divide into without leaving a remainder.

To find the LCM of the denominators, we generally list the multiples of each number until we find a common one. Another method is to use the prime factorization of each number and then multiply each factor the greatest number of times it appears in any of the numbers. For instance, for the numbers 6 and 8:

• Prime factorization of 6 is 2 × 3
• Prime factorization of 8 is 2 × 2 × 2

The LCM will have to include both prime factors 2 and 3. Since the highest power of 2 present in either number is 2³, we use 2³ for the LCM. We also include 3 (from the prime factorization of 6) and multiply them together, which yields 2³ × 3 = 8 × 3 = 24. When variables are involved, such as in the given example with denominators 6x and 8x, we also consider the variable as part of the LCM. The LCM in this case is therefore 24x.

Combining Fractions with a Common Denominator

When we have two or more fractions that we wish to combine, having a common denominator is key. The common denominator serves as a shared base that allows the fractions to be added or subtracted easily. Once the LCM is identified, you can adjust each fraction so they all have this same denominator.

For example, if we are working with the fractions \(\frac{x-3}{6x}\) and \(\frac{x+4}{8x}\), we first convert them to have the common denominator of 24x by finding equivalent fractions that maintain the same value. To do this, we multiply the numerator and the denominator of each fraction by the factor required to reach the LCM:

• \(\frac{x-3}{6x} \times \frac{4}{4} = \frac{4(x-3)}{24x}\)
• \(\frac{x+4}{8x} \times \frac{3}{3} = \frac{3(x+4)}{24x}\)

Once both fractions have the common denominator, they can be added together directly across the numerator because the denominators match, resulting in a single fraction with that common denominator.

Simplifying Algebraic Expressions in Fractions

After combining fractions with a common denominator, we often need to simplify the resulting algebraic expression. Simplifying involves reducing the expression to its simplest form by combining like terms and eliminating common factors from the numerator and the denominator.

In the example \(\frac{4x-12+3x+12}{24x}\), the like terms in the numerator are combined, which in this case are 4x and 3x, and then -12 and +12 cancel each other out. The new expression \(\frac{7x}{24x}\) is thus obtained. The next step in simplification is to look for common factors in both the numerator and the denominator. In this instance, we observe that both the numerator and the denominator share 'x' as a common factor. Cancelling out the common variable 'x', we are left with \(\frac{7}{24}\), which is the simplified form of the original complex fraction and cannot be reduced further.
Simplifying algebraic expressions requires careful observation and a systematic approach to ensure that all like terms and common factors are correctly identified and simplified.