Problem 58 - Algebra 1: Concepts and Skills

Question

Find the opposite of the number. $$ 7.5 $$

Step-by-Step Solution

Answer

The opposite of \(7.5\) is \(-7.5\).

Step 1: Identify the given number

The number given in the problem is \(7.5\).

Step 2: Compute the opposite

The opposite of a number is just the negative of that number. So, simply append a negative sign to the number.

Key Concepts

Negative Numbers

Negative numbers are the numbers that are less than zero. They are found on the left side of zero on the number line. Negative numbers are often used in various real-world situations, like representing temperatures below freezing, or owing money in a bank account.
For example, if you have a balance of -$20, it means you owe $20. So, these numbers are crucial to understand. Notably, each positive number has a corresponding negative number. This pair concept leads us to opposites.
To find the opposite of a number, you simply change its sign:

• For a positive number, the opposite is negative.
• For a negative number, the opposite is positive.

So, the opposite of 7.5 is -7.5.

Absolute Value

Absolute value measures how far a number is from zero on the number line, without considering its direction. It's always a non-negative number.
Consider the absolute value of -3, which is 3, since -3 is 3 units away from zero. Similarly, the absolute value of 7.5 is 7.5 itself.
The notation for absolute value uses vertical bars. So, if \( x \) is any real number, the absolute value is written as \( |x| \). Therefore:

• \( |-5| = 5 \)
• \( |7.5| = 7.5 \)

Absolute value is particularly useful in understanding real-world scenarios like distance, where only the magnitude matters, not the direction.

Number Line

A number line is a visual tool that represents numbers in a straight, horizontal layout. It's a simple yet powerful way to understand the order and size of numbers, both positive and negative.
The center of a number line is zero. Positive numbers are located to the right of zero, while negative numbers are to the left.

• For example,
  • -3 is left of -1
  • 4 comes after 2 on the right side

Using a number line helps us to visualize concepts like greater than, less than, and equal to. It also aids in easy computation of opposites; just find a number on the line, and its mirror image across zero is its opposite. So, if you find 7.5 on the number line, simply look directly the same distance on the other side to find -7.5.