Math Expression: Find The Missing Step

Hey math whizzes! Today, we're diving into a super fun problem that’ll test your skills with order of operations. We’ve got an expression, $[(-10+2)-1]+(2+3)$, and we need to figure out how to get from Step 1 to Step 2. It’s like solving a puzzle, guys, and each step brings us closer to the final answer. We’ll break down the whole process, make sure you totally get what’s going on, and reveal the missing piece of the puzzle. So, buckle up, and let’s get this math party started!

Understanding the Order of Operations

Before we jump into our specific problem, let's quickly chat about the order of operations. You know, that handy rule that tells us which part of a math problem to solve first. We usually remember it with the acronym PEMDAS or BODMAS. It stands for:

• Parentheses (or Brackets)
• Exponents (or Orders)
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

This order is super important because it ensures that everyone gets the same answer for the same problem. Without it, math would be a total free-for-all! In our problem, we have parentheses and brackets, so we’ll be focusing on solving those parts first. We need to be really careful with our signs, especially when we're dealing with negative numbers. It’s easy to make a small slip-up there, but by going step-by-step, we can avoid any confusion. Remember, inside the innermost parentheses or brackets, you perform the operations first. Then you work your way outwards. This systematic approach is key to mastering these kinds of expressions. Let’s keep this in mind as we tackle our puzzle.

Deconstructing the Problem: From Expression to Step 2

Alright, let's look at our original expression: $[(-10+2)-1]+(2+3)$. We're given Step 2, which is $-9+2+3$. The question is, what happened between the original expression and Step 2? We need to find the expression that belongs in Step 1.

Let's focus on the first part of the expression inside the square brackets: $[(-10+2)-1]$.

• Inside the parentheses, we have $(-10+2)$. When you add a positive number to a negative number, you find the difference between their absolute values and take the sign of the number with the larger absolute value. So, $-10+2$ equals $-8$.
• Now our expression inside the brackets looks like $[-8-1]$.
• Performing the subtraction inside the brackets, $-8-1$ equals $-9$.

So, the entire first part of the expression $[(-10+2)-1]$ simplifies to $-9$.

Now let's look at the second part of the expression: $(2+3)$.

• This is straightforward addition: $2+3$ equals $5$.

Putting it all together, the original expression $[(-10+2)-1]+(2+3)$ simplifies to $-9+5$.

Now, let's compare this to what we see in Step 2: $-9+2+3$. It looks like the first part of our calculation, $(-10+2)-1$, correctly simplified to $-9$. However, the second part, $(2+3)$, which equals $5$, seems to have been rewritten as $2+3$ in Step 2, instead of being combined into a single number. This is a common way to show intermediate steps in some problem-solving methods, where you might simplify parts of an expression without fully combining them immediately.

So, if Step 2 is $-9+2+3$, it means that the first bracketed part $[(-10+2)-1]$ must have been simplified to $-9$, and the second part $(2+3)$ was left as $2+3$. This implies that Step 1 must have represented the expression where the simplification of the first part has occurred, but the second part is still separate. The most direct representation of this would be $[-9]+(2+3)$ or something very close to it that leads directly into $-9+2+3$. Let's re-examine the options provided to see which one fits this transition perfectly.

Evaluating the Options

We're looking for the expression in Step 1 that logically leads to Step 2 ($-9+2+3$) after some calculation. Step 2 shows that the first part of the expression resulted in $-9$, and the second part $(2+3)$ was kept as is. Let's see which option, when simplified partially, gives us this.

• Option A: $[-10+-1+2]+(2+3)$

  • Let's simplify inside the brackets: $-10+-1+2$. Adding $-10$ and $-1$ gives $-11$. Then, $-11+2$ equals $-9$. So, this option simplifies to $[-9]+(2+3)$. This is exactly what we need! If we remove the brackets, we get $-9+2+3$, which is Step 2. This looks like a strong contender, guys.

• Option B: $[-8-1]+(2+3)$

  • Simplifying inside the brackets: $-8-1$ equals $-9$. So, this option simplifies to $[-9]+(2+3)$. Removing the brackets gives $-9+2+3$, which is Step 2. This also looks like a strong contender. Let's think about how we got to $-8-1$. It came from $(-10+2)-1$. So, $[-8-1]$ is a valid intermediate step from $[(-10+2)-1]$. This means both A and B are plausible as they both lead to $[-9]+(2+3)$ which then becomes $-9+2+3$.

• Option C: $[-10+1]+(2+3)$

  • Simplifying inside the brackets: $-10+1$ equals $-9$. So, this option simplifies to $[-9]+(2+3)$. Removing the brackets gives $-9+2+3$, which is Step 2. Another strong contender. However, how did we get $-10+1$? This doesn't seem to directly follow from $(-10+2)-1$ in a single logical step based on PEMDAS unless there was a typo in the original expression or the options.

• Option D: $[8+1]+(2+3)$

  • Simplifying inside the brackets: $8+1$ equals $9$. So, this option simplifies to $[9]+(2+3)$. This would lead to $9+2+3$, which is not $-9+2+3$. So, option D is definitely out.

Pinpointing the Correct Step 1

We need to be really precise here. The original expression is $[(-10+2)-1]+(2+3)$.

Let's trace the simplification process step-by-step:

1. Inside the innermost parentheses: $(-10+2) = -8$.
2. The expression becomes: $[-8-1]+(2+3)$.
3. Now, simplify inside the brackets: $-8-1 = -9$.
4. The expression becomes: $[-9]+(2+3)$.
5. If we remove the brackets and keep the addition sign: $-9+2+3$. This is exactly Step 2!

So, the question is asking what expression is missing from Step 1, implying that Step 1 is the result of an initial simplification. Let's re-read the question carefully: "The three steps below were used to find the value of the expression... Step 1: ? Step 2: $-9+2+3$ Step 3: $-7+3$ Which expression is missing from Step 1?"

This means Step 1 is a stage in the calculation. Based on our breakdown, the stage before we get to $-9+2+3$ (Step 2) is when the initial calculation within the brackets has been done, leaving $-9$, and the second part $(2+3)$ is still separate.

Let's re-evaluate options A, B, and C based on the original expression $[(-10+2)-1]+(2+3)$:

• Option A: $[-10+-1+2]+(2+3)$

  • This expression, if performed, would result in $[-9]+(2+3)$, which leads to $-9+2+3$. The internal calculation $[-10+-1+2]$ is a reordering and regrouping of $[-10+2-1]$. This is mathematically valid. So, Step 1 could be this expression.

• Option B: $[-8-1]+(2+3)$

  • This expression is directly obtained from $[(-10+2)-1]+(2+3)$ by first calculating $(-10+2)=-8$. So, $[(-10+2)-1]$ becomes $[-8-1]$. This seems to be the most direct intermediate step from the original expression before the final simplification within the brackets.

• Option C: $[-10+1]+(2+3)$

  • This option implies $-10+1$, which equals $-9$. This does not directly come from $(-10+2)-1$ without altering the numbers significantly. For instance, if it was $[-10+2-1]$, then summing $-10$ and $2$ gives $-8$. Then $-8-1$ gives $-9$. Option C has a +1 instead of +2-1. This seems incorrect.

Let's reconsider the transition. The original expression is $[(-10+2)-1]+(2+3)$.

Step 1 should represent a state where part of the original expression is simplified. Step 2 is $-9+2+3$. Step 3 is $-7+3$ (which is $-4$, and Step 2 is $-9+5 = -4$. So Step 3 follows Step 2).

The transition from the original expression to Step 2 involves:

1. Simplifying $(-10+2)-1$ to $-9$.
2. Leaving $(2+3)$ as $2+3$. (Note: $(2+3)=5$. So Step 2 is $-9+5$. The expression $-9+2+3$ evaluates to $-9+5$.)

So, Step 1 must be an expression that evaluates to $-9+2+3$ or an expression that becomes $-9+2+3$ after a simple step (like removing brackets).

Let's look at the options again:

• A. $[-10+-1+2]+(2+3)$

  • This evaluates to $[-9]+(2+3)$. Removing brackets gives $-9+2+3$. This is exactly Step 2. The internal expression $-10+-1+2$ is a valid rearrangement of $-10+2-1$.

• B. $[-8-1]+(2+3)$

  • This evaluates to $[-9]+(2+3)$. Removing brackets gives $-9+2+3$. This is exactly Step 2. The internal expression $-8-1$ comes from $(-10+2)-1$. This is the most direct calculation within the brackets.

• C. $[-10+1]+(2+3)$

  • This evaluates to $[-9]+(2+3)$. Removing brackets gives $-9+2+3$. This is exactly Step 2. However, the internal expression $-10+1$ doesn't directly follow from $(-10+2)-1$.

The question asks for the missing expression from Step 1. This implies Step 1 is a result of simplifying the original expression. The structure of the original expression is $[(a)-b]+(c+d)$.

When we calculate $(-10+2)$, we get $-8$. So, the expression becomes $[-8-1]+(2+3)$. This is Option B.

If Step 1 is $[-8-1]+(2+3)$, then the next step is to calculate $-8-1 = -9$, giving $[-9]+(2+3)$. Removing brackets gives $-9+2+3$, which is Step 2.

So, Option B seems to be the most direct and logical representation of the expression after the very first simplification inside the brackets.

Let's consider the possibility that Step 1 is already Step 2, but written slightly differently, or that Step 1 is what precedes Step 2.

If Step 1 is the expression that simplifies to Step 2, and Step 2 is $-9+2+3$, then Step 1 should be something that, when you do one more calculation, you get $-9+2+3$. This usually means simplifying brackets.

Let's look at the provided answer choices again in light of how they directly relate to the original expression $[(-10+2)-1]+(2+3)$.

• Option B: $[-8-1]+(2+3)$

  • This is derived from $[(-10+2)-1]+(2+3)$ by calculating $(-10+2) = -8$. So, Step 1 could be seen as the expression with this first part simplified.

• Option A: $[-10+-1+2]+(2+3)$

  • This requires reordering inside the brackets. While mathematically valid, Option B is a more direct computational step.

• Option C: $[-10+1]+(2+3)$

  • This requires changing the numbers, which isn't a simplification step.

Given the sequence implies simplification, Option B, $[-8-1]+(2+3)$, represents the state after the innermost parentheses $(-10+2)$ have been evaluated to $-8$. The next logical step would be to evaluate $-8-1$ within the brackets.

If Step 1 is $[-8-1]+(2+3)$, then the next calculation is $-8-1 = -9$. This gives us $[-9]+(2+3)$. Then, removing the brackets (or simply combining the terms since it's just addition) gives $-9+2+3$, which is Step 2.

Therefore, Option B is the missing expression for Step 1 because it represents the state of the expression after the first calculation within the parentheses has been performed, leading directly to the form seen in Step 2 after one more calculation.

The Final Answer

After carefully breaking down the original expression and analyzing each step, we can confidently determine the missing expression for Step 1. The original expression is $[(-10+2)-1]+(2+3)$.

1. First, we simplify the innermost parentheses: $(-10+2) = -8$.
2. This leads to the expression: $[-8-1]+(2+3)$. This is our Step 1.
3. Next, we simplify within the square brackets: $-8-1 = -9$. The expression becomes $[-9]+(2+3)$.
4. Removing the brackets and performing the addition yields: $-9+2+3$. This is Step 2.
5. Finally, we complete the addition in Step 2: $-9+2+3 = -7+3 = -4$. (Note: The provided Step 3 is $-7+3$, which follows from $-9+2+3$ by first calculating $-9+2 = -7$. So, the steps are consistent).

Based on this sequence, the expression missing from Step 1 is B. $[-8-1]+(2+3)$. It’s the logical intermediate step that directly transforms into Step 2. Pretty neat, right?

Remember, math is all about following the rules and taking it one step at a time. Keep practicing, and you'll become a master problem-solver in no time! Happy calculating, everyone!