Problem 25 - Intermediate Algebra

Question

Perform the operations. See Example 2 . $$ -3.3-(-3.3) $$

Step-by-Step Solution

Answer

The result is 0.

Step 1: Understanding the Problem

We are given the expression \[-3.3 - (-3.3)\] and need to perform the operations to simplify it.

Step 2: Rewrite the Expression

Remember that subtracting a negative number is the same as adding its positive. The expression can therefore be rewritten as:\[-3.3 + 3.3\]

Step 3: Add the Numbers

Add \[-3.3\] and \[3.3\].When you perform this operation, you get:\[0\]

Step 4: Verify Your Solution

To verify, notice that adding a number to its negative gives zero, confirming that \[-3.3 + 3.3 = 0\].

Key Concepts

Understanding Negative Numbers

Negative numbers are an essential concept in mathematics. They are numbers less than zero, represented by a minus sign (-). You can think of negative numbers as a way of describing a deficit or a loss. For example, if you owe $5, your balance would be -5. In number lines, negative numbers are placed to the left of zero. This helps visually understand their size in relation to positive numbers.

When dealing with negative numbers in expressions or equations, always pay attention to the operation involved. In subtraction, removing a negative is like adding its positive counterpart. This is because subtracting a negative flips it to positive, often simplifying the problem or expression. Keeping this principle in mind can make operations involving negative numbers much clearer and straightforward.

Basic Number Operations

Number operations form the basis of arithmetic and mathematical problem solving. These include addition, subtraction, multiplication, and division, each with its own rules and special properties.

Let's focus on addition and subtraction first:

• **Addition**: When you add numbers, you combine their values. Commutative and associative properties govern addition, meaning the order doesn’t affect the result.
• **Subtraction**: Subtracting involves finding the difference between numbers. Structurally, it’s the inverse of addition.

A useful tip is remembering that subtraction can sometimes be rephrased as adding a negative. In the expression a - b, you can think of it as a + (-b), useful for simplifying and better understanding equations.

When subtracting a negative number, as with -3.3 - (-3.3), translate this directly to addition: -3.3 + 3.3. This approach is key in simplifying and solving these types of problems.

Verifying Solutions

Verifying solutions is a crucial step in mathematics. It ensures that your solution is correct and reliable. This process involves reassessing the steps you took to arrive at your solution to catch any errors.

In the context of the expression -3.3 - (-3.3), the simplest way to verify the outcome is by recognizing that adding a number and its opposite results in zero. This rule stems from the additive inverse property, which states for any number a, a + (-a) = 0. Applying this logic confirms the accuracy of the solution.
Verification also highlights understanding. By going through the verification, you reinforce why rules like the additive inverse exist and how they apply in various scenarios. In mathematics, verifying isn’t only about ensuring accuracy—it deepens comprehension and builds confidence in handling similar operations.