Problem 58 - Intermediate Algebra

Question

Perform the operations and simplify the result when possible. See Example 6 $$\frac{x-1}{4 x-24}-\frac{3 x-2}{5 x-30}$$

Step-by-Step Solution

Answer

The simplified result is \( \frac{-7x+3}{20(x-6)} \).

Step 1: Identify the denominators

In the given expression \( \frac{x-1}{4x-24}-\frac{3x-2}{5x-30} \), identify the denominators. The first denominator is \( 4x-24 \) and the second is \( 5x-30 \).

Step 2: Factor the denominators

Factor each denominator by taking out the greatest common factor. For \( 4x-24 \), factor out a 4: \( 4(x-6) \). For \( 5x-30 \), factor out a 5: \( 5(x-6) \).

Step 3: Find the common denominator

The common denominator of \( 4(x-6) \) and \( 5(x-6) \) is \( 20(x-6) \). You achieve this by multiplying the individual denominators \( 4(x-6) \) and \( 5(x-6) \).

Step 4: Rewrite fractions with the common denominator

Rewrite each fraction to have the common denominator \( 20(x-6) \).- Multiply the numerator and denominator of \( \frac{x-1}{4(x-6)} \) by 5 to get \( \frac{5(x-1)}{20(x-6)} \).- Multiply the numerator and denominator of \( \frac{3x-2}{5(x-6)} \) by 4 to get \( \frac{4(3x-2)}{20(x-6)} \).

Step 5: Subtract the numerators

Subtract the numerators of the fractions: \( 5(x-1) - 4(3x-2) \).Let us distribute and simplify:- \( 5(x-1) = 5x - 5 \)- \( 4(3x-2) = 12x - 8 \)This results in: \( 5x - 5 - (12x - 8) = 5x - 5 - 12x + 8 = -7x + 3 \).

Step 6: Write the final expression

Write the new fraction using the simplified numerator: \( \frac{-7x+3}{20(x-6)} \).

Key Concepts

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. Just like regular fractions, rational expressions can be simplified, added, or subtracted. The key difference is that, unlike simple numerical fractions, we have polynomials involved, making things a tad more complex but still manageable.
When handling rational expressions:

• Ensure that the denominator is not zero, as division by zero is undefined.
• Always try to simplify by factoring both the numerator and the denominator.

These simple guidelines help soften the complexities of dealing with rational expressions, making operations like simplification, addition, or subtraction more approachable.

Factoring

Factoring is the process of breaking down a polynomial into simpler components known as factors, which when multiplied together give back the original polynomial. In our exercise, we factor the denominators to find a common base which helps in performing operations on rational expressions.

To factor,

• Look for the greatest common factor (GCF) in the terms of the polynomial.
• For example, in the expression \(4x - 24\), the GCF is 4, so \(4x - 24\) can be factored as \(4(x-6)\).
• Similarly, the expression \(5x-30\) can be factored by its GCF, 5, resulting in \(5(x-6)\).

Factoring helps to reveal the structure of the polynomial, simplifying further operations and making it easier to find a common denominator.

Common Denominator

Finding a common denominator is crucial when adding or subtracting rational expressions or fractions. The common denominator is essentially a shared multiple of the original denominators that allows you to perform operations across different rational expressions.

To find the common denominator:

• Factor each denominator.
• Identify common factors.
• Multiply the unique factors. Here, \(4(x-6)\) and \(5(x-6)\) share \(x-6\), so the common denominator becomes \(20(x-6)\).

Using a common denominator allows the numerators to be subtracted directly, setting the stage for further simplifications.

Subtracting Fractions

Subtracting fractions, whether they have polynomials or numbers, follows the same basic rules:

• Ensure a common denominator.
• Adjust the numerators to reflect this shared base.

In our exercise, we adapted each fraction to the common denominator \(20(x-6)\), allowing the numerators to be rewritten and subtracted easily.

After obtaining a common denominator:

• Subtract the numerators: \(5(x-1) - 4(3x-2)\), which simplifies to \(-7x + 3\) after distributing and combining like terms.
• Finally, write this resulting expression as a single fraction over the common denominator.

This methodical approach ensures clarity and precision when subtracting rational expressions.