Problem 24 - Elementary Algebra

Question

The number 5 is the opposite of \(-5\). Explain why.

Step-by-Step Solution

Answer

5 is the opposite of -5 because it is the same distance from zero but in the opposite direction.

Step 1: Understanding Opposites

Opposite numbers are numbers that are the same distance from zero on the number line but in opposite directions. For example, the opposite of a positive number is its corresponding negative number, and vice versa.

Step 2: Identify the Given Number

The given number is \(-5\). This is a negative number located five units to the left of zero on the number line.

Step 3: Locate the Opposite on the Number Line

To find the opposite of \(-5\), identify which number is the same distance from zero but in the opposite direction. This number will be five units to the right of zero, which is \(5\).

Step 4: Conclusion

\(5\) is the opposite of \(-5\) because it is the same distance from zero as \(-5\), but on the other side of the number line.

Key Concepts

Understanding the Number Line

A number line is a visual representation of numbers in a straight line. Each point on the line corresponds to a number. The number line generally extends infinitely in both the positive and negative directions.

Zero is at the center of the number line and acts as a reference point. Positive numbers are to the right of zero, while negative numbers are to the left.

• A number line helps visualize the concept of opposite numbers.
• It also helps in understanding the distance between numbers.

Positive and Negative Numbers

Every number on the number line has a unique position that defines whether it is positive or negative.

Positive numbers are greater than zero and are found to the right of zero on the number line. Negative numbers are less than zero and are found to the left.

• A positive number and its corresponding negative number are called opposites because they are the same distance from zero, but in different directions.
• For example, the number 5 is opposite to -5.

Distance from Zero

The distance of a number from zero is also known as its absolute value. This distance is always a positive value, as it simply denotes how far a number is from zero, regardless of direction.

• For a positive number like 5, the distance from zero is 5 units.
• For a negative number like -5, the distance from zero is also 5 units.

Understanding distance from zero helps in identifying opposite numbers. Both 5 and -5 are 5 units away from zero, making them opposites on the number line.