Problem 16 - Beginning and Intermediate Algebra

Question

Subtract. See Examples 1 through 5 $$ -4-(-16) $$

Step-by-Step Solution

Answer

The result of \(-4 - (-16)\) is 12.

Step 1: Identify the Operation

The problem presented is a subtraction problem of the form \(-4 - (-16)\). We need to understand that subtracting a negative number is equivalent to adding a positive number.

Step 2: Rewrite with Addition

Transform the subtraction operation by removing the negative sign from the second number and changing subtraction to addition: \(-4 - (-16)\) becomes \(-4 + 16\). This is because subtracting a negative is the same as adding its positive equivalent.

Step 3: Perform the Addition

Now, perform the simple addition calculation: \(-4 + 16\). Start from -4 and move 16 units to the right on the number line, which yields a result of 12.

Key Concepts

Integer Subtraction

Subtraction of integers involves taking one integer away from another. Essentially, you are finding the difference between two numbers. However, when one or both numbers are negative, the process can appear confusing. To begin, always identify which number you are subtracting from (known as the minuend) and which number you are subtracting (called the subtrahend). In our exercise, we are subtracting (-16) from -4, meaning -4 is our minuend and -16 is our subtrahend.

• If both numbers are positive, regular subtraction rules apply.
• If both are negative, consider their absolute values to determine the difference logically.
• Subtracting a negative number changes the operation to addition, simplifying many problems like ours: -4 - (-16) = -4 + 16.

This understanding is vital for handling any integer subtraction problem effectively.

Negative Number Rules

Negative numbers can seem tricky at first. But, like positive numbers, they follow specific rules that, once mastered, make calculations straightforward. Here are some key rules for dealing with negative numbers:

• Adding a negative number is equivalent to subtracting the corresponding positive number. For instance, 4 + (-3) is the same as 4 - 3.
• Subtracting a negative number is the same as adding the positive equivalent. Therefore, -5 - (-7) becomes -5 + 7.
• Multiplying or dividing two negative numbers results in a positive number. However, multiplying or dividing a negative and a positive number yields a negative result.

Understanding these rules will help simplify and clarify subtraction problems involving negative numbers.

Number Line Addition

One of the simplest ways to visualize adding and subtracting integers is using a number line. The number line helps make complex operations more intuitive. Imagine a straight line marked with numbers at equal intervals. In the context of our problem, start at the point representing the first number, which is -4. Since subtraction of a negative number becomes addition, move to the right 16 units. Here's how the process appears on a number line:

• Start at -4: Locate -4 on your number line.
• Move right 16 spaces: Because you are 'adding' 16.
• Result: You land on 12, showing that -4 - (-16) equals 12.

Using a number line often makes abstract arithmetic operations concrete and less confusing, especially with negative numbers.