Problem 105 - Algebra A Combined Function

Question

Each calculation below is incorrect. Find the error and correct it. See the Concept Check in this section. $$ -10+(-12) \stackrel{?}{=}-120 $$

Step-by-Step Solution

Answer

The correct answer is \(-22\), not \(-120\).

Step 1: Understanding the operation

The original calculation is \(-10 + (-12)\). This expression involves adding two negative numbers, not multiplication.

Step 2: Identify the operation error

The given equation suggests the result \(-120\), which indicates a possible misuse of multiplication instead of proper addition of integers.

Step 3: Correct the operation

When adding \(-10 + (-12)\), the signs remain negative, so the calculation is equivalent to \(-(10 + 12)\). This equals \(-22\).

Step 4: Verify the calculation

Recheck the operation by adding \(-10\) and \(-12\) on a number line if needed: Starting from \(-10\), move 12 units further in the negative direction to reach \(-22\).

Step 5: Confirm the solution

The correct result of the calculation \(-10 + (-12)\) is \(-22\), not \(-120\).

Key Concepts

Negative Numbers

Adding negative numbers can seem tricky at first, but it is straightforward once you understand the basic rules. Think of negative numbers as steps going down a staircase. When you add more negative numbers, you are going further down.

In the expression \(-10 + (-12)\), both \(-10\) and \(-12\) are negative. So, you're starting at \(-10\) on the number line and then taking 12 more steps in the negative direction. Together, these steps sum up to \(-22\), which means you are 22 steps below zero.

Here's how you can think about it in simple terms:

• Adding a negative is like subtracting a positive. So, \(-10 + (-12)\) simplifies to \(-(10 + 12)\).
• Keep track of your steps on the number line. If you keep going negative, your total becomes more negative.

Operation Error

Operation errors arise when there is a mix-up in how we are supposed to work with the numbers. In many problems, these errors come from mixing addition with multiplication or subtraction.

In the original example, the error was thinking the addition of negatives turned into multiplication, which isn't the case. When you see a plus sign between numbers, even if they're negative, you're still adding. The mistake of obtaining \(-120\) indicates multiplying instead, which is not needed here.

Always be clear about which operation the problem requires:

• Read expressions carefully to avoid using the wrong operation.
• Remember, combining negatives through addition keeps numbers negative instead of jumping to another operation like multiplication.

Number Line

A number line is a fantastic visual tool that helps clarify integer addition, especially with negative numbers. Imagine a horizontal line with zero in the middle. All negative numbers are to the left, and positive numbers are to the right.

To solve \(-10 + (-12)\) using a number line, start at \(-10\) and move 12 units further to the left. This lands you at \(-22\), showing the accurate sum.

Using a number line is beneficial because:

• It offers a clear way to see where numbers are added or subtracted without confusion.
• You can see visually how adding negatives makes you go further left, making the sum more negative.

The number line is especially valuable when learning to add and subtract because it gives a concrete sense of how numbers move and change direction.