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Chapter 7: Problem 7

Perform the indicated operation. Simplify, if possible. \(\frac{1}{6 r}-\frac{3}{8 r}\)

Short Answer

Expert verified
The simplified expression is \[ \frac{3x + 5}{12} \].

Step by step solution

01

Identify the common denominator

Notice that both fractions have the same denominator of 12.
02

Combine the numerators

Since the denominators are the same, we can combine the fractions by adding the numerators: \[ \frac{x + (2x + 5)}{12} \].
03

Simplify the numerator

Combine the like terms in the numerator: \[ x + 2x + 5 = 3x + 5 \]. So the fraction becomes \[ \frac{3x + 5}{12} \].
04

Final answer

The expression \[ \frac{x}{12}+\frac{2 x+5}{12} \] simplifies to \[ \frac{3x + 5}{12} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

common denominator
When adding fractions, finding a common denominator is crucial. A denominator is the number below the fraction bar. To add fractions easily, ensure they have the same denominator.
In our exercise, both fractions already have the same denominator, 12. This step simplifies our work because we do not need to adjust the denominators.
By having a common denominator, we can combine the fractions directly.
combining numerators
Once you have identified a common denominator, the next step is to combine the numerators. The numerator is the number above the fraction bar.
In the given exercise, the numerators are \( x \) and \( 2x + 5 \). Because the denominators are the same, we add these numerators directly:
\[ \frac{x + (2x + 5)}{12} \]

This step is simpler when the fractions share the same denominator. Just remember to keep the denominator unchanged while you add the numerators.
simplifying algebraic expressions
The final step involves simplifying the algebraic expression in the numerator. We have
\[ x + (2x + 5) = x + 2x + 5 \]
By combining like terms, we get \( 3x + 5 \).
Our fraction now becomes:
\[ \frac{3x + 5}{12} \]

Simplifying algebraic expressions means combining all like terms. Look for terms that have the same variable parts to combine them easily.
This makes our final solution neat and simpler to understand.

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