Problem 70 - Beginning and Intermediate Algebra

Question

Find each additive inverse or opposite. See Example \(9 .\) $$ -\frac{1}{4} $$

Step-by-Step Solution

Answer

The additive inverse of \(-\frac{1}{4}\) is \(\frac{1}{4}\).

Step 1: Understand the Concept of Additive Inverse

The additive inverse or opposite of a number is what you add to that number to get zero. If you have a number \(a\), its additive inverse is \(-a\). This holds true for both integers and fractions.

Step 2: Identify the Original Number

We start with the given number, which is \(-\frac{1}{4}\). This is our original number for which we need to find the additive inverse.

Step 3: Determine the Additive Inverse

To find the additive inverse of \(-\frac{1}{4}\), we take the negative of that number, which means we change the sign. So, the additive inverse is \(-(-\frac{1}{4})\).

Step 4: Simplify the Expression

When you have a double negative, it becomes positive. Therefore, \(-(-\frac{1}{4}) = \frac{1}{4}\). This means that the additive inverse of \(-\frac{1}{4}\) is \(\frac{1}{4}\).

Step 5: Verify the Result

To ensure that \(\frac{1}{4}\) is indeed the additive inverse, add \(-\frac{1}{4}\) and \(\frac{1}{4}\). The sum is \(-\frac{1}{4} + \frac{1}{4} = 0\). Since the sum is zero, our solution is correct.

Key Concepts

Understanding Fractions

Fractions are a way of expressing numbers that are not whole. They consist of a numerator and a denominator. The numerator is the top number and represents how many parts you have. The denominator is the bottom number and tells you how many parts make up a whole. For example, in the fraction \( \frac{1}{4} \), 1 is the numerator, and 4 is the denominator, meaning you have 1 part out of 4 total parts.

Fractions can have different properties. They can be positive or negative, just like whole numbers. Negative fractions have a negative sign either in the numerator, denominator, or in front of the fraction itself. In the exercise, we dealt with the fraction \(-\frac{1}{4}\), where the negative sign indicates the entire fraction is less than zero.

Fractions can be simplified or expressed differently. Understanding fractions is fundamental to solving many math problems, especially when dealing with operations like addition, subtraction, and finding inverses.

Exploring Negative Numbers

Negative numbers are numbers less than zero. They are typically represented by placing a minus sign in front of a number. In mathematics, negative numbers have many uses, including in real-life contexts like debts or temperatures below zero.

When dealing with negative numbers, certain rules apply. One of the most important rules is that when you multiply or divide two negatives, you get a positive. This is why in the solution, the additive inverse of \(-\frac{1}{4}\) is positive \(\frac{1}{4}\). When you negate a negative number, it becomes positive.

• Negative number combined with a positive number of equal magnitude results in zero.
• Negating a negative gives a positive.

Thus, understanding negative numbers is crucial for working with equations and algebraic expressions involving both negatives and positives.

Basic Algebra Concepts

Algebra is a branch of mathematics that uses letters and symbols to represent numbers and quantities in formulas and equations. Basic algebra helps in understanding the relationships between numbers and finding unknown values.

One essential algebraic concept is the idea of additive inverses. Additive inverses are numbers which, when added together, yield zero. For any number \(a\), its additive inverse is \(-a\). When we add these two, we get \(a + (-a) = 0\). This principle is crucial when solving equations as it allows you to isolate variables and find solutions easily.

Another key concept is knowing how to manipulate equations by performing operations such as addition, subtraction, multiplication, and division. These operations help to simplify expressions and solve equations efficiently.

• Additive inverse helps to simplify equations.
• Basic operations allow you to rearrange and solve equations effectively.

Grasping these algebraic fundamentals is important as it builds the foundation for more complex mathematical problem-solving.