Problem 59 - Algebra 1: Concepts and Skills

Question

Determine 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

Yes, the statement 'If you subtract a positive number from a negative number, the result is always a negative number' is true. The number line and the subtraction rule both validate this statement.

Step 1: Define the Operation

Subtracting a positive number from a negative number can be visually understood with the help of a number line. The process actively involves moving to the left on the number line. The subtraction can be alternatively written as 'negative number plus negative of positive number.'

Step 2: Applying the Scenario on Number Line

Let's take a scenario where -4 (a negative number) is being subtracted from 2 (a positive number) on a number line. So, the operation will become -4 - 2. Now, starting at -4 on the number line, for subtracting 2 we will move 2 times to the left which brings us to the -6 spot on the number line. As the result lies on the negative side, the statement seems true for this scenario.

Step 3: Validating with Subtraction Rule

Subtracting a positive number from a negative number can be rewritten as addition of two negative numbers as per the rule a - b = a + (-b). So as per this rule, -4 - 2 will become -4 + (-2) = -6. Considering the intuitive principle that addition of two negative numbers will always be a negative, we can validate the statement.

Key Concepts

Number Line

A number line is a visual tool used to understand numbers and their operations. It is a horizontal line where:

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

This setup allows for an easy visualization of operations like addition and subtraction. When we subtract using a number line, we move:

• To the left if we subtracted a positive number.
• To the right if we subtracted a negative number.

Using a number line can simplify understanding integer operations, making it especially useful for visual learners.

Positive and Negative Numbers

Positive numbers are greater than zero and are typically represented without a sign or with a '+' sign. Examples include 1, 2, and 3. Negative numbers are less than zero and are represented with a '-' sign such as -1, -2, and -3. These concepts are important when performing arithmetic operations.

When dealing with both positive and negative numbers, it's helpful to keep in mind:

• Addition of a positive number increases the value, moving the number to the right on the number line.
• Addition of a negative number decreases the value, shifting it to the left on the number line.
• Subtracting a positive number is similar to adding a negative number, resulting in a movement to the left on the number line.
• Subtracting a negative number is like adding a positive number, causing a rightward movement on the number line.

Understanding these properties helps in accurately solving arithmetic problems involving both types of numbers.

Subtraction Rule

The subtraction rule provides a helpful way to think about subtraction and simplify problems involving negative and positive numbers. It states that:

• Subtracting a number is the same as adding its opposite. For instance, instead of thinking about subtracting 2, you can think of adding -2.
• This rule helps to transform any subtraction operation: \( a - b = a + (-b) \).

Let's consider a practical example:

Suppose we have the expression \(-4 - 2\). Using the subtraction rule, this can be rewritten as \(-4 + (-2)\). Both operations will end up at the same spot on the number line. By thinking of subtraction as adding a negative number, calculations can become more intuitive. This technique can be particularly beneficial in handling negative integers while ensuring the correct result is consistently reached.