Problem 65 - Algebra 1

Question

Find the opposite of the number. $$ 7.9 $$

Step-by-Step Solution

Answer

The opposite of 7.9 is -7.9

Step 1: Identify the given number

The given number is 7.9.

Step 2: Find the opposite number

The opposite of a number is obtained by changing its sign. If the number is positive, the opposite is negative and vice versa. Here, since the given number 7.9 is positive, its opposite is -7.9.

Key Concepts

Positive and Negative Numbers

Understanding the concept of positive and negative numbers is fundamental in mathematics, particularly when dealing with the concept of opposites. Positive numbers are the numbers you frequently use in daily life for counting, such as 1, 2, and 3, which lie to the right of zero on the number line. Negative numbers, on the other hand, are less intuitive since they represent quantities less than zero, like -1, -2, -3, and they lie to the left of zero on the number line.

Negative numbers are used in various real-world contexts, including temperatures below freezing, elevations below sea level, or bank balances indicating debt. The opposite of a positive number is its negative counterpart, which is the same distance from zero on the number line, but in the opposite direction. Similarly, the opposite of a negative number is a positive number. For instance, the opposite of +7.9 is -7.9, and the opposite of -7.9 is +7.9.

Algebra Basics

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. It's a powerful tool that allows for generalizing arithmetic operations and solving a wide range of problems. When we talk about the opposite of a number in algebra, we're often referring to its additive inverse.

The additive inverse of any number is a number which, when added to the original number, will result in zero. This concept is also described by the equation \( a + (-a) = 0 \), where \( a \) is any number, and \( -a \) is its opposite. Whenever you're asked to find the opposite of a number in algebra, you simply give the number with the opposite sign without changing its magnitude.

Number Properties

Number properties are the basic rules that numbers follow. These include properties such as the commutative, associative, distributive properties, and properties of zero and one. Specifically relevant here is the understanding of how additive inverses work, as it relates to the property of opposites within numbers.

One such important property states that every number has a unique opposite, which ensures the existence of only one additive inverse for every real number. Another key property is the symmetry of opposites; mathematically expressed, if \( b \) is the opposite of \( a \), then \( a \) is also the opposite of \( b \). Comprehending these properties allows students to better visualize and solve algebraic problems that involve opposites and inverses.