Problem 34 - Elementary and Intermediate Algebra

Question

Subtract and simplify the result, if possible. \(\frac{5 x+8}{3 x+15}-\frac{3 x-2}{3 x+15}\)

Step-by-Step Solution

Answer

The simplified result is \(\frac{2}{3}\).

Step 1: Identify the Common Denominator

Both fractions have the common denominator \(3x + 15\). This means we can combine the fractions by subtracting their numerators while keeping the same denominator.

Step 2: Subtract the Numerators

Subtract the second numerator from the first: \[(5x + 8) - (3x - 2) = 5x + 8 - 3x + 2.\]This simplifies to:\[2x + 10.\]

Step 3: Write the Combined Numerator Over the Common Denominator

Place the new numerator over the common denominator:\[\frac{2x + 10}{3x + 15}.\]

Step 4: Simplify the Fraction

Factor out the greatest common factor from the numerator and the denominator. The numerator, \(2x + 10\), can be factored as \(2(x + 5)\). The denominator, \(3x + 15\), can be factored as \(3(x + 5)\).

Step 5: Cancel the Common Factor

Since the factor \((x + 5)\) is common to both the numerator and the denominator, it can be cancelled out, resulting in: \[\frac{2}{3}.\]

Key Concepts

Subtracting Fractions

Subtracting fractions may seem tricky at first, but it becomes simple once you grasp a few key ideas. When dealing with subtraction, you need to focus on the numerators - the top numbers in the fraction.

• Keep the denominator (the bottom number) the same.
• Subtract the second fraction's numerator from the first fraction's numerator.

For example, in the original exercise, the expression was \(\frac{5x+8}{3x+15} - \frac{3x-2}{3x+15}\). You simply subtract \(3x-2\) from \(5x+8\). Here's how it breaks down:- Subtraction of \((5x + 8) - (3x - 2)\)- Simplify this subtraction to get \(2x + 10\).In this way, you keep everything neat, just by focusing on the numerators!

Common Denominator

A common denominator is vital when subtracting fractions, as it enables the fractions to be combined easily.

• Fractions must have the same denominator to be subtracted directly.
• If denominators differ, you'd need to find a common one, typically the least common multiple (LCM).

Luckily, in the provided exercise, \(3x+15\) was already the common denominator for both fractions. This streamlined the process because you didn't have to adjust the denominators first, which is often the trickiest part of handling fractions!

Fraction Simplification

Simplifying fractions is an important step that makes the results cleaner and easier to understand. After you've performed subtraction and written the new fraction, check if simplification is possible.Here's how:

• Look for common factors between the numerator and the denominator.
• Factor these out to make the fraction simpler.

For instance, the expression \(\frac{2x + 10}{3x + 15}\) can be simplified because the numerator and the denominator both have common factors.- Factor 2 from \(2x + 10\): \(2(x + 5)\).- Factor 3 from \(3x + 15\): \(3(x + 5)\).Recognizing and cancelling out \((x + 5)\) simplifies this to \(\frac{2}{3}\)!

Factoring

Factoring is a technique used to simplify expressions by finding terms that multiply together to give the original expression. This process is crucial, particularly when simplifying fractions.In the original problem:

• Both the numerator, \(2x + 10\), and the denominator, \(3x + 15\), are factored to reveal common terms.
• For \(2x + 10\), recognize that it can be rewritten by factoring out a 2, resulting in \(2(x + 5)\).
• For \(3x + 15\), factor out a 3, resulting in \(3(x + 5)\).

Once these common terms, here \((x + 5)\), are identified, they can be cancelled from both the numerator and denominator. This simplification makes the fraction far more manageable, such that \(\frac{2}{3}\) is easily obtained.