Problem 52 - Prealgebra

Question

Give the opposite of each of the following numbers. $$-5$$

Step-by-Step Solution

Answer

The opposite of -5 is 5.

Step 1: Understanding Opposites

The opposite of a number is what you must add to a number to get zero. It is effectively the negative of a number if it is positive, and the positive of a number if it is negative.

Step 2: Identify the Given Number

We are given the number \(-5\). This is a negative number.

Step 3: Find the Opposite

To determine the opposite of \(-5\), we change its sign. Thus, the opposite is \(5\) because \(-5 + 5 = 0\).

Key Concepts

Negative Numbers

A negative number is any number less than zero. It is usually indicated by a minus sign (-) before the number. Negative numbers are used to represent quantities that are less than nothing such as debts or temperatures below freezing. When you subtract a positive number from zero, you get a negative number.

For example:

• If you owe \(5\) dollars, your financial situation can be represented as \(-5\) dollars.
• When the temperature falls \(10\) degrees below zero, it is expressed as \(-10\) degrees.

Using negative numbers might seem tricky at first. But once you understand that they imply moving to the left on a number line, the concept becomes easier to grasp.

Number Line

A number line is a visual representation where numbers are arranged in order along a straight line. The center of a number line is usually the point zero (0).

As you move right from zero, the numbers increase and are positive. Moving to the left of zero, you find negative numbers. On a number line:

• Zero is the neutral starting point.
• Positive numbers are on the right side.
• Negative numbers are located on the left side.

The direction you move on a number line determines whether you are dealing with positive or negative numbers. Keeping this layout in mind helps visualize operations like addition, subtraction, and finding opposites of numbers.

Understanding the number line sets a foundation for exploring more complex number systems like integers and real numbers.

Absolute Value

The absolute value of a number is its distance from zero on a number line, without considering its sign. This means absolute value is always a non-negative number.

To find the absolute value of a number, use the notation \(|x|\) where \(x\) is the number.

Here are some examples:

• The absolute value of \(-5\) is \(|-5| = 5\), because it is 5 units away from zero.
• Similarly, the absolute value of \(7\) is \(|7| = 7\), since it is also 7 units away from zero.

The concept of absolute value is crucial when comparing distances or differences regardless of direction, as distances are considered equal, whether you go left or right from the origin.