Solving The Numerical Expression: −3−(−2) + 3−[4−5·(−1)]

by Blender 57 views

Hey guys! Today, we're diving into a fun little math problem. We need to figure out the numerical value of the expression −3−(−2) + 3−[4−5·(−1)]. Sounds intimidating? Don't worry, we'll break it down step by step. We have four options to choose from: A) 1, B) 2, C) 3, and D) 4. But we're not just picking an answer; we're going to justify it by showing all the steps. So, grab your thinking caps, and let's get started!

Breaking Down the Expression Step-by-Step

To solve this, we'll follow the golden rule of math: the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Let's dive in and make sure we get this right, guys.

Step 1: Tackle the Innermost Parentheses

First up, we need to deal with what's inside the brackets: [4−5·(−1)]. Within these brackets, we have both subtraction and multiplication. According to PEMDAS, multiplication comes before subtraction. So, let's handle the 5·(−1) part first.

  • Key Concept: Multiplying a positive number by a negative number results in a negative number. In this case, 5 multiplied by -1 is -5. So, we have:

    4 − 5 ⋅ (−1) = 4 − (−5)

Step 2: Dealing with the Double Negative

Now we have 4 − (−5). Here's a neat trick: subtracting a negative number is the same as adding its positive counterpart. Think of it like this: you're taking away a debt, which is like gaining something. So, −(−5) becomes +5.

  • Key Concept: Subtracting a negative is equivalent to addition. So,

    4 − (−5) = 4 + 5 = 9

Step 3: Rewriting the Expression

Now that we've simplified the innermost parentheses, let's rewrite the entire expression. We've figured out that [4−5·(−1)] is equal to 9. So, our expression now looks like this:

−3 − (−2) + 3 − 9

Step 4: Addressing the Remaining Operations

We're left with subtraction and addition. Remember, addition and subtraction are on the same level in the order of operations, so we perform them from left to right.

Step 4.1: −3 − (−2)

Just like before, we have a double negative situation. Subtracting −2 is the same as adding 2. So:

  • Key Concept: Again, subtracting a negative turns into addition.

    −3 − (−2) = −3 + 2

Step 4.2: −3 + 2

Now we're adding a negative number and a positive number. Think of it as owing $3 and then gaining $2. You'll still be in debt, but by a smaller amount. The result is -1.

  • Key Concept: Adding numbers with different signs involves finding the difference and using the sign of the larger number.

    −3 + 2 = −1

Step 5: Continuing the Calculation

Let's plug that back into our expression. We now have:

−1 + 3 − 9

Step 5.1: −1 + 3

Adding -1 and 3 is like having $3 and spending $1. You'll have $2 left.

  • Key Concept: Adding numbers with different signs again. 3 is larger than 1, so the result will be positive.

    −1 + 3 = 2

Step 6: The Final Step

Our expression is now super simple:

2 − 9

Step 6.1: 2 − 9

This is like having $2 and needing to pay $9. You're short $7. So, the result is -7.

  • Key Concept: Subtracting a larger number from a smaller number results in a negative number.

    2 − 9 = −7

Conclusion: The Correct Answer and Why

Alright, guys, we've gone through each step meticulously, and we've arrived at our final answer: -7. However, looking back at the original options, we see A) 1, B) 2, C) 3, and D) 4. None of these match our calculated result of -7.

Wait, What Happened?

It seems there might have been a mistake in the original options provided. Our step-by-step solution clearly shows that the numerical value of the expression −3−(−2) + 3−[4−5·(−1)] is -7. It’s essential in mathematics to double-check our work and ensure that each step is logically sound. We followed the order of operations (PEMDAS) diligently, and our calculations are accurate. So, if we were to provide the correct answer based on our work, it would be None of the above or -7.

Importance of Showing Your Work

This exercise highlights why showing your work is super important in math. By laying out each step, we can easily identify any errors or discrepancies. In this case, we confirmed that our process was correct, which allowed us to confidently state that the provided options were incorrect. It also helps anyone following along to understand the logic and reasoning behind each step, making the learning process more effective and transparent.

Final Thoughts

So, there you have it, guys! We tackled a numerical expression, navigated through parentheses and negative signs, and arrived at the correct answer. Even though the provided options didn't include our result, we stood by our calculations and demonstrated the importance of a step-by-step approach in mathematics. Keep practicing, and these types of problems will become second nature. Great job, and keep up the awesome work!