Problem 33 - Beginning and Intermediate Algebra

Question

Perform the operation. See Example 3. Subtract \(-5\) from \(8 .\)

Step-by-Step Solution

Answer

The result is 13.

Step 1: Understand the Operation

The operation requires subtracting (-5) from 8. This can be written as: 8 - (-5).

Step 2: Apply the Rule of Subtraction for Negative Numbers

When subtracting a negative number, you add the absolute value of that number. So, 8 - (-5) becomes 8 + 5.

Step 3: Perform the Addition

Now perform the addition: 8 + 5 = 13.

Key Concepts

Integer Operations

Integer operations include addition, subtraction, multiplication, and division involving whole numbers, which can be both positive and negative. When performing operations with integers, it's important to follow specific rules to get correct results.

For instance, when you add two positive integers together, the result is greater than either of the two numbers. Adding two negative integers will produce a result that is less than either of them.
For example:

• Adding integers: -3 + (-2) = -5
• Subtracting integers: 4 - 3 = 1
• Multiplying integers: (-4) × 2 = -8
• Dividing integers: (-6) / 2 = -3

Subtraction, particularly, can be tricky because it also involves the rule of subtracting negative numbers, which actually becomes an addition problem. So, understanding and mastering integer operations are crucial foundations for solving many math problems efficiently.

Absolute Value

The absolute value of a number indicates its distance from zero on the number line, regardless of direction. This means it is always a non-negative number, as it represents the size or magnitude of the number without any reference to its sign.

For example, the absolute value of -5 is 5, denoted as \(|-5| = 5\). Similarly, the absolute value of 7 is also 7, written as \(|7| = 7\).
This concept is particularly useful when you are dealing with operations involving negative numbers because it simplifies calculations by focusing on the magnitude rather than the direction.
In integer operations, especially in subtraction: when you "subtract a negative," you essentially need to add the absolute value of that number. This is why we change 8 - (-5) to 8 + 5 in the original problem.

Negative Numbers

Negative numbers are numbers less than zero, representing opposite direction on the number line compared to positive numbers. They can be a little confusing initially, especially in subtraction operations, since the rules differ from positive numbers.

Negative numbers are crucial in various real-life scenarios, such as temperatures below zero, elevations below sea level, or financial debts.
Understanding how to manipulate them can also give insights into controlling gains and losses in these situations.

• A key property of negative numbers is that when you add a negative number to a positive, the negative may decrease the overall value, depending on which has a greater absolute value.
• For example, subtracting a negative number changes the operation into addition. Hence, subtracting -5 from 8 becomes an addition problem: 8 + 5, which is easier to solve.
• This rule simplifies the subtraction operation, turning what might have been a complication into something straightforward.

Remember, any time you subtract a negative, you're essentially adding the number's absolute value, bringing us back to the simplicity of basic addition operations.