Problem 36 - Algebra for College Students

Question

Perform the following operations with real numbers. $$-32.6-(-9.8)$$

Step-by-Step Solution

Answer

-22.8

Step 1: Identify the Operation and the Numbers

The given expression is \(-32.6 - (-9.8)\). This involves subtracting a negative number from a negative number.

Step 2: Simplify the Double Negative

When you subtract a negative number, it is equivalent to adding the positive form of that number. Convert the expression \(-32.6 - (-9.8)\) to \(-32.6 + 9.8\).

Step 3: Perform the Addition

Now, perform the operation \(-32.6 + 9.8\). When you add a positive number (9.8) to a negative number (-32.6), the result will be negative but closer to zero. Calculate: \(-32.6 + 9.8 = -22.8\).

Key Concepts

Subtracting Negative Numbers

When we come across the operation of subtracting a negative number, it can be a bit counterintuitive at first. Think of negative numbers as having direction, much like temperature below zero or an elevator moving downwards. The negative sign indicates a move in the opposite direction of the positive numbers. So, when you subtract a negative, you are essentially removing the effect of that negative value.

• Subtracting \(-(-9.8)\) is like saying "Let's move in the positive direction by 9.8."
• This changes the operation from subtraction to addition.

Understanding this can clear up a lot of confusion, especially when solving problems involving multiple operations with real numbers.

Addition with Negative Numbers

Adding a negative number is another concept that can be a little tricky. When you add a negative number, you are effectively decreasing the total, moving the value further away from zeros on the number line. Consider an elevator analogy: if you're at the 5th floor and you go down 2 floors, you end up on the 3rd floor.

• Imagine \(-32.6 + 9.8\), where you add a positive to a negative.
• The positive number acts as a counterforce, reducing the overall negative impact.
• Think of it like summing debts and credits: each positive (credit) offsets part of the negative (debt).

This step-by-step approach to adding negative and positive numbers will build a better understanding of how real-number operations work.

Double Negative Simplification

Simplifying double negatives is an essential part of mastering operations with real numbers. When you encounter two negative signs in a mathematical expression, it's important to remember that they transform into a positive. Just as in language where a double negative implies a positive, the same applies in math.

• For example, \(-(-9.8)\) equals \(+9.8\).
• A practical approach is to cancel out the negatives, making it simpler to perform arithmetic operations.
• This concept is very useful in algebra, calculus, and even in day-to-day problem-solving.

By mastering the technique of turning these double negatives into positives, solving expressions becomes straightforward and less error-prone.