Problem 35 - Elementary and Intermediate Algebra

Question

Change the sign. (Find the opposite.) $$ 7 $$

Step-by-Step Solution

Answer

-7

Step 1: Understand the Concept of Opposite Numbers

Opposite numbers are pairs of numbers that are the same distance from zero on the number line but in opposite directions. For any number, its opposite is obtained by changing its sign.

Step 2: Identify the Given Number

The given number in the exercise is 7.

Step 3: Change the Sign

To find the opposite of 7, change its sign from positive to negative.

Step 4: Write the Result

The opposite of 7 is -7.

Key Concepts

number line

A number line is a straight, horizontal line used to display numbers at regular intervals. It helps to visualize numbers and their relationships. The center of the number line is zero. Numbers to the right of zero are positive, while numbers to the left are negative. Number lines are useful because they make it easy to see how numbers are spaced out and how they relate to each other.

For example, if you have a number line with only whole numbers, it might look like this:

-3, -2, -1, 0, 1, 2, 3. Each of these numbers is evenly spaced, making it easy to understand their values.

One important aspect of the number line is understanding where opposite numbers are located. Opposites are equal distances from zero but on opposite sides. For example, the opposite of 3 is -3, and they are both three units away from zero, just in different directions.

positive and negative numbers

Positive and negative numbers are fundamental concepts in mathematics. Positive numbers are greater than zero and are located to the right of zero on the number line. Examples include 1, 2, 3, and so on. They represent things like quantities, distances, or heights that are above or beyond a starting point.

Negative numbers, on the other hand, are less than zero and are found to the left of zero on the number line. Examples include -1, -2, -3, and so on. These numbers can represent things like debts, temperatures below freezing, or depths below sea level.

Understanding positive and negative numbers is crucial for solving problems that involve changes in value, directions, or measurements. For instance, if you move left on the number line, you are going into negative territory. If you move right, you enter positive territory.

absolute value

The absolute value of a number is the distance between that number and zero on the number line, regardless of direction. It's always a non-negative number because distance can't be negative. The absolute value is written using vertical bars, like this: \(|a|\).

For example:
• \(|7| = 7\)
• \(|-7| = 7\)

This shows that both 7 and -7 are seven units away from zero, even though they are on opposite sides of the number line.

When solving problems, the concept of absolute value helps to simplify expressions and equations. It can also be used to compare numerical differences without worrying about direction. For example, \(|a - b|\) can tell you the difference between two numbers a and b, regardless of which one is larger.