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Chapter 1: Problem 446

Subtract. $$ -\frac{3}{7}-\left(-\frac{5}{7}\right) $$

Short Answer

Expert verified
\( \frac{2}{7} \)

Step by step solution

01

Identify the Expression

The expression to be simplified is \( - \frac{3}{7} - \big( - \frac{5}{7} \big) \).
02

Understand the Subtraction of a Negative Number

Subtracting a negative number is equivalent to adding its positive counterpart. Therefore, \( - ( - \frac{5}{7} ) \) becomes \( + \frac{5}{7} \).
03

Rewrite the Expression

Rewrite the expression by replacing the subtraction of the negative number: \( - \frac{3}{7} + \frac{5}{7} \).
04

Perform the Addition

Since the denominators are the same, subtract the numerators: \( -3 + 5 = 2 \). Thus, \( - \frac{3}{7} + \frac{5}{7} = \frac{2}{7} \).

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Key Concepts

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

Integers
Integers are whole numbers, both positive and negative, including zero. Thus, they don't include any fractional or decimal part. Examples of integers are -3, -2, -1, 0, 1, 2, 3, and so on.
When performing operations with integers, especially within fractions, it's essential to keep track of their positive or negative sign. Understanding integers forms the basis of more complex mathematics operations such as those involving fractions.
For instance, in our original expression, we see both positive and negative integers in the numerators of the fractions. Therefore, understanding integers and how to operate with them is crucial for solving such exercises.
Negative Numbers
Negative numbers are those less than zero. They are represented with a minus sign (-). For example, -1, -2, -3 are all negative numbers.
One important rule when dealing with negative numbers in operations is understanding subtraction. Subtracting a negative number is the same as adding its positive counterpart. This can be confusing but is crucial.
In our exercise, you can see this rule in action. When we have \(-\frac{3}{7} - \left( -\frac{5}{7} \right)\), it is equivalent to \(-\frac{3}{7} + \frac{5}{7}\). This changes the operation from subtraction to addition, by flipping the sign of the second term. This simple step simplifies the problem significantly.
Adding Fractions
Adding fractions involves a few straightforward steps, particularly when the denominators are the same.
Here鈥檚 the summarized process:
  • Ensure the denominators are the same, if not find a common denominator.
  • Add the numerators together.
  • Place the result over the common denominator.
  • Simplify the fraction if necessary.
In the provided exercise, the initial step is often the most crucial. Because the denominators are the same (both are 7), we can skip the step of finding a common denominator and go straight to adding the numerators.
We rewrite the expression as \(-\frac{3}{7} + \frac{5}{7}\). The next step is simple: add the numerators. The calculation becomes \(-3 + 5 = 2\). Place this result over the common denominator to achieve our final result: \(\frac{2}{7}\).

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