Problem 60 - Prealgebra

Question

Find each difference. $$-5-5$$

Step-by-Step Solution

Answer

The difference is -10.

Step 1: Identify the Terms

The expression given is \(-5 - 5\), where both terms are negative."

Step 2: Understand the Meaning of Subtraction

Subtracting a positive number, like 5, from a negative number, like -5, involves moving further in the negative direction on a number line.

Step 3: Combine the Terms

When we subtract 5 from -5, it's equivalent to adding -5 more to -5. Therefore, the equation is \(-5 - 5 = -10\).

Key Concepts

Negative Numbers

Negative numbers are values that are less than zero. They are important in mathematics because they help represent losses, debts, or decreases. For example, if you owe someone money, you can think of that debt as a negative value. Negative numbers have unique rules, especially when it comes to operations like addition and subtraction.
They are often denoted with a minus sign "-" in front of the number. When performing operations with negative numbers, it's important to remember these rules:

• Adding a negative number is the same as subtracting its positive counterpart. For example: \(-3 + (-2) = -3 - 2\).
• Subtracting a negative number is the same as adding the positive version. For example: \(5 - (-3) = 5 + 3\).

Understanding these rules will help you perform operations with negative numbers more confidently.

Number Line

A number line is a simple visual representation that can help you understand counting, addition, and subtraction with both positive and negative numbers. Picture it as a straight line where numbers are laid out in order from negative on the left to positive on the right.
On a number line:

• Zero is usually placed in the center.
• Negative numbers are to the left of zero.
• Positive numbers are to the right of zero.

When subtracting numbers using a number line, imagine your starting point at the first number. If you want to subtract a positive number, move left that many spaces. In our example, for the expression \(-5 - 5\), you start at \(-5\) and move 5 spaces left because you are subtracting 5. This ultimately lands you at \(-10\), demonstrating why \(-5 - 5\) equals \(-10\). This visual method can make understanding the concepts of negative numbers and subtraction much less intimidating.

Combining Terms

Combining terms is a useful technique in arithmetic that simplifies expressions by putting together similar parts. It's often used in simplifying algebraic expressions but is just as handy for simplifying arithmetic problems with negative numbers.
When you combine terms, you look at the numbers and their operators (+ or -) to determine how they add or subtract from each other. In the example \(-5 - 5\), both terms are negative, leading you down the number line.
To combine these terms:

• Recognize that subtracting a positive number is like adding its negative. So \(-5 - 5\) becomes \(-5 + (-5)\).
• Add the absolute values of the numbers and keep the negative sign, resulting in \(-10\).

This process is fundamental in arithmetic involving integers, as it ensures you're accurately reflecting the magnitude and direction (positive or negative) of each value in calculations.