Problem 83 - Algebra 1

Question

Decide whether the statement is true or false. Use the subtraction rule or a number line to support your answer. If you subtract a positive number from a negative number, the result is always a negative number.

Step-by-Step Solution

Answer

True, if you subtract a positive number from a negative number, the result is certainly a negative number.

Step 1: Understanding the Statement

It's stated that if a positive number is subtracted from a negative number, the result will always be a negative number. Take a general negative number, for instance -5 and a generic positive number, let's say 3. So we need to evaluate -5 - 3.

Step 2: Subtracting on a Number Line

On a number line there is action of 'take away' represented by subtraction; it involves moving to the left of the initial number by the number of places equivalent to the subtracted number. Starting from -5 you move 3 places to the left to land on -8.

Step 3: Conclusion on the Result

After moving 3 places to the left from -5 on the number line, we land on -8, which is manifestly a negative number. Thus, the procedure validates the given statement that states - 'If you subtract a positive number from a negative number, the result is always a negative number.'

Key Concepts

Subtraction Rule

Understanding how subtraction operates with both positive and negative numbers is fundamental to grasping mathematics. The subtraction rule simplifies this understanding. When we subtract any number, we are essentially finding the difference between two quantities. If the number you are subtracting is positive, you're moving to the left on the number line. Conversely, subtracting a negative number (which is like adding a positive number of the same absolute value) will move you to the right.

Take, for example, subtracting a positive number from a negative one. In numeric terms, let's consider \( -7 - 3 \). According to the rule, you start at -7 on the number line and move 3 units to the left, thus increasing the negative value. The rule effectively tells us that \( -7 - 3 = -10 \), resulting in a more negative number. Remember, any time you move left on the number line, the value decreases. Subtraction of a positive number from any number will therefore make it either less positive (if starting with a positive number) or more negative (if starting with a negative number).

Number Line

A number line is a visual representation of numbers placed on a straight horizontal line. It is a powerful educational tool that illustrates the concept of subtraction clearly. On this line, numbers to the right are positive and increase in value, while numbers to the left are negative and decrease in value. A number line can make abstract concepts more tangible, especially when we talk about negative numbers and subtracting.

For instance, if we start at -4 and need to subtract 2, we would physically find -4 on the line and move 2 units to the left, landing on -6 (\( -4 - 2 = -6 \)). This visual process demonstrates how subtraction moves a value along the number line and reinforces understanding of the subtraction rule and the relationship between positive and negative numbers.

Negative Numbers

In the context of mathematics, negative numbers represent values less than zero. They are crucial for describing quantities that exist in the opposite direction from positive numbers, such as debt or temperature below zero. The concept of negative numbers is intimately linked to the number line, where they appear on the left side of zero.

Understanding subtraction with negative numbers can be tricky, but remember this — subtracting a larger number from a smaller number will always yield a negative result. Therefore, when a positive number is subtracted from a negative number, it results in a more negative number (i.e., a number that is further to the left on the number line). In simpler terms, negative numbers become larger in magnitude but smaller in value the further they are from zero.