Problem 62 - Elementary and Intermediate Algebra

Question

Subtract. $$ -4-(-7) $$

Step-by-Step Solution

Answer

-4 - (-7) = 3

Step 1: Identify the Operation

First, identify that the problem is asking to subtract \( -7 \) from \( -4 \).

Step 2: Rewrite the Expression

Rewrite the subtraction of a negative number as an addition: \( -4 - (-7) \) becomes \( -4 + 7 \).

Step 3: Perform the Addition

Add the two numbers: \( -4 + 7 = 3 \). Note that adding a positive number to a negative number reduces the negative number's absolute value.

Key Concepts

Integer Operations

Integer operations include addition, subtraction, multiplication, and division of whole numbers, both positive and negative. These operations follow specific rules that help us determine the result. For clarity:

• Adding two positive integers always results in a positive integer.
• Adding two negative integers results in a negative integer.
• Subtracting a positive integer from a negative integer is like adding a negative integer, which results in a more negative number.
• Subtracting a negative integer from another negative integer can be tricky, which we'll explain in the next sections.

Understanding these rules is crucial for mastering integer operations.

Rewrite Subtraction as Addition

When faced with the subtraction of negative numbers, we can make the problem simpler by rewriting it as an addition. This is because subtracting a negative is equivalent to adding a positive. For example, in the problem: \( -4 - (-7) \), we can rewrite it as \( -4 + 7 \).

This technique allows us to handle the operation more easily. When we rewrite subtraction as addition, it helps to think of the two negative signs next to each other as transforming into a positive sign:

• The first negative sign indicates the operation (subtraction).
• The second negative sign belongs to the number being subtracted.

So, \( -(-7) \) turns into \( +7 \).Transforming the problem this way makes it a lot easier to solve.

Addition of Positive and Negative Numbers

After rewriting the subtraction as an addition, solving the problem requires understanding how to add positive and negative numbers. Using our example, \( -4 + 7 \), we need to add a positive seven to a negative four.

Here's the process:

• Start from -4 on the number line.
• Since we are adding 7, move 7 units to the right.
• You will land on 3.

When adding a positive number to a negative number, you reduce the absolute value of the negative number. Essentially, you are combining a loss (negative) with a gain (positive). If the positive number is larger, the result is positive. If the negative number is larger, the result stays negative.

Thus, in \( -4 + 7 = 3 \), because 7 is larger than 4, we end up with a positive 3. This concept is fundamental in understanding operations with positive and negative numbers.