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Chapter 6: Problem 41

Perform the indicated operations. Simplify when possible $$ \frac{3}{t}-\frac{6}{-t} $$

Short Answer

Expert verified
\frac{9}{t}

Step by step solution

01

Recognize the Negative Sign in the Denominator

Identify that \( -t \) in the second fraction can be rewritten in terms of \( t \). Thus, \( \frac{6}{-t} = -\frac{6}{t} \).
02

Rewrite the Expression

Once recognizing that \( \frac{6}{-t} = -\frac{6}{t} \), rewrite the expression as \( \frac{3}{t} - (-\frac{6}{t}) \).
03

Simplify the Expression

Combine the fractions: \( \frac{3}{t} - (-\frac{6}{t}) = \frac{3}{t} + \frac{6}{t} = \frac{3+6}{t} = \frac{9}{t} \).

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Key Concepts

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

Understanding Negative Signs in Expressions
When working with negative signs in algebraic expressions, it鈥檚 vital to understand their impact. In our exercise, we have a negative sign in the denominator of the fraction \[ \frac{6}{-t} \].
This means we can rewrite this fraction. Negative signs in fractions can be tricky, but here's a simple way to handle them:
  • When you encounter a fraction with a negative denominator, rewrite the fraction by moving the negative sign to the front: \[ \frac{6}{-t} = -\frac{6}{t} \].
  • This helps to simplify the fraction and makes further operations easier.
By moving the negative sign out front, we make the fraction more manageable for combining with others. Remember, understanding the role of negative signs in expressions is key to simplifying and solving algebraic equations efficiently.
Simplifying Fractions
Simplifying fractions is a fundamental skill in algebra. In the given exercise, our goal is to simplify the expression \[ \frac{3}{t} - \frac{6}{-t} \]. After addressing the negative sign, we rewrite it as \[ \frac{3}{t} - \big( -\frac{6}{t} \big) \].
Here鈥檚 a step-by-step approach to simplify such fractions:
  • Identify and Rewrite: First, rewrite any fractions with negative denominators.
  • Combine like fractions: Ensure the fractions have the same denominator, making them ready to combine (add or subtract).
  • Combine the terms: Add or subtract the fractions directly. Since the denominators are the same, combine the numerators and keep the common denominator.
In this exercise, we rewrite and simplify as follows:
\[ \frac{3}{t} + \frac{6}{t} = \frac{3+6}{t} = \frac{9}{t} \]
This simplified fraction is our final answer and shows how a seemingly complex set of terms can be simplified into a single fraction.
Combining Like Terms
Combining like terms is an essential concept when simplifying algebraic expressions. In our exercise, we dealt with fractions having the same denominator:
\[ \frac{3}{t} + \frac{6}{t} \].
When the denominators are identical, it becomes straightforward to combine the terms:
  • Same Denominator: Ensure that the fractions have the same denominator.
  • Add/Subtract the Numerators: Combine the numerators while retaining the common denominator.
  • Simplify: If possible, simplify the resulting fraction.
So, we have our expression:
\[ \frac{3}{t} + \frac{6}{t} \]
This becomes:
\[ \frac{3+6}{t} = \frac{9}{t} \]
Combining like terms simplifies the expression, making it more manageable and easier to understand. Understanding how to combine like terms is crucial in making complex algebraic equations simpler and more solvable.

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