Problem 30 - Algebra for College Students

Question

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ \frac{5}{6 x}-\frac{3}{10 x} $$

Step-by-Step Solution

Answer

\(\frac{8}{15x}\)

Step 1: Find the Least Common Denominator (LCD)

The denominators of the expressions are \(6x\) and \(10x\). First, we identify the least common multiple of 6 and 10, which is 30. Therefore, the least common denominator for these two terms is \(30x\).

Step 2: Rewrite each Fraction with the LCD

Convert each fraction to have the common denominator \(30x\). To do this, convert \(\frac{5}{6x}\) to \(\frac{5 \times 5}{30x}\) since \(6x \times 5 = 30x\). Similarly, convert \(\frac{3}{10x}\) to \(\frac{3 \times 3}{30x}\) since \(10x \times 3 = 30x\).

Step 3: Subtract the Fractions

Now that the fractions have a common denominator, subtract the numerators: \(\frac{25}{30x} - \frac{9}{30x} = \frac{25 - 9}{30x} = \frac{16}{30x}\).

Step 4: Simplify the Resulting Fraction

Simplify \(\frac{16}{30x}\) by dividing the numerator and the denominator by their greatest common divisor, which is 2. This gives \(\frac{8}{15x}\).

Step 5: Conclusion: Write the Simplified Expression

The simplified form of the expression \(\frac{5}{6x} - \frac{3}{10x}\) is \(\frac{8}{15x}\).

Key Concepts

Least Common Denominator (LCD)

Finding the Least Common Denominator (LCD) is a crucial step when dealing with rational expressions. It allows us to compare or perform operations, such as addition and subtraction, on fractions with different denominators. The LCD is essentially the smallest common multiple of the denominators. For instance, in the expression \(\frac{5}{6x} - \frac{3}{10x}\), both fractions have denominators with the variable \(x\). We first focus on the numerical parts: 6 and 10.
Multiplying these directly would give us a denominator of 60, which is not minimal. So, instead, we find the least common multiple (LCM), which for 6 and 10 is 30 (since 30 is the smallest number that both 6 and 10 can divide into without leaving a remainder).

• Steps to Find the LCD:
• List the multiples of each number.
• Identify the smallest multiple common to both lists, which is the LCM.
• Incorporate any variables. Here, with the common \(x\), the LCD becomes \(30x\).

Once you have the LCD, you can rewrite each fraction to have this common denominator, simplifying further operations.

Simplifying Fractions

Simplifying fractions is a process of reducing a fraction to its simplest form, where the numerator and denominator have no common factors other than 1. This is important not just for aesthetic appeal, but also for ease of computation in further steps. Let’s look back at the expression \(\frac{16}{30x}\), which is the result from subtracting \(\frac{5}{6x}\) and \(\frac{3}{10x}\).
To simplify \(\frac{16}{30x}\), you perform the following steps:

• Find the greatest common divisor (GCD) of 16 and 30.
• Divide both the numerator and the denominator by this GCD.

For 16 and 30, the GCD is 2. So, dividing \(\frac{16}{30x}\) by 2 gives \(\frac{8}{15x}\).
In mathematical terms, \(\frac{16 \div 2}{30x \div 2} = \frac{8}{15x}\). By doing this, you ensure your answer is in its simplest form, making it neat and transparent for anyone reading your work.

Subtracting Rational Expressions

Subtracting rational expressions involves several steps: aligning denominators, performing the subtraction, and simplifying the result. With our expression, your first task was finding that common denominator, which we've established is \(30x\).
Once both fractions were rewritten to have this common denominator:

• \(\frac{5}{6x}\) became \(\frac{25}{30x}\) because \(6x \times 5 = 30x\).
• \(\frac{3}{10x}\) became \(\frac{9}{30x}\) because \(10x \times 3 = 30x\).

Now, with the same denominators, subtract the numerators directly:
\(\frac{25}{30x} - \frac{9}{30x} = \frac{16}{30x}\)
This step-by-step subtraction is straightforward because the denominators match. Finally, simplifying \(\frac{16}{30x}\) into \(\frac{8}{15x}\) completes the operation.
Remember, ensuring the denominators are the same before subtracting allows for a simple "top minus top" approach, making rational expressions easier to handle and solve.