Problem 22 - Algebra A Combined Function

Question

Subtract. \(-\frac{4}{7}-\left(-\frac{1}{7}\right)\)

Step-by-Step Solution

Answer

The result is \(-\frac{3}{7}\).

Step 1: Analyze the Expression

We need to perform the subtraction operation given by \(-\frac{4}{7} - \left(-\frac{1}{7}\right)\). Pay attention to the subtraction of a negative fraction which requires changing signs.

Step 2: Change Subtraction to Addition

Subtraction of a negative number can be changed to an addition of its positive equivalent. Thus, \(-\frac{4}{7} - \left(-\frac{1}{7}\right) = -\frac{4}{7} + \frac{1}{7}\).

Step 3: Combine the Fractions

Since the fractions have a common denominator of 7, we can directly combine them: \(-\frac{4}{7} + \frac{1}{7} = \frac{-4+1}{7}\).

Step 4: Simplify the Result

Simplify the result by performing the addition in the numerator: \(\frac{-4+1}{7} = \frac{-3}{7}\).

Key Concepts

Negative Numbers

When dealing with negative numbers, it's important to understand how they affect operations like addition and subtraction. A negative number is any number less than zero, represented by a minus sign (-). Negative numbers appear in various scenarios, such as temperatures below freezing or debts below zero.

In arithmetic operations, subtracting a negative number is equivalent to adding its positive counterpart. This is because subtraction of a negative actually means removing a debt or deficiency, which is akin to adding a gain. Consider the expression \(-a - (-b)\), which becomes \(-a + b\). Therefore, if you see \(-(-b)\), it's simply the positive \(+b\). This key insight helps simplify expressions and solve problems involving negative numbers effectively.

Common Denominators

To perform operations on fractions, especially addition and subtraction, it’s crucial to have a common denominator. This is because fractions are a representation of division, showing parts of a whole.

A common denominator is a shared multiple of the denominators of two or more fractions, allowing us to add or subtract the numerators directly. For example, in the expression \(-\frac{4}{7} + \frac{1}{7}\), the denominator 7 is already common. Thus, we can easily perform the fraction operation by keeping the denominator the same and operating on the numerators. This simplifies calculations and ensures accuracy, as differing denominators will give incorrect results.

• Ensure fractions have the same denominator before adding or subtracting them.
• Use the least common multiple (LCM) of the denominators if necessary to convert different denominators.

Fraction Operations

Fraction operations involve basic arithmetic with fractions, allowing us to add, subtract, multiply, and divide them. To correctly perform these operations, it is vital to understand the rules specific to fractions.

For subtraction, such as in \(-\frac{4}{7} + \frac{1}{7}\), once you have a common denominator, focus on the numerators. Here, the operation involves adding -4 and 1, resulting in -3. The outcome can be expressed as \(\frac{-3}{7}\).

• Keep the denominator unchanged when adding or subtracting fractions with the same denominator.
• Simplify fractions whenever possible to express the simplest form.
• Recognize when subtracting fractions that involve negative values may need converting between subtraction and addition operations.

Practice these strategies to become adept at handling fraction operations, ensuring any fraction-based calculations are performed smoothly and correctly.