Problem 28 - Beginning and Intermediate Algebra

Question

Add. See Examples I through 7. $$ 9.2+(-11.4) $$

Step-by-Step Solution

Answer

The sum is \(-2.2\).

Step 1: Identify the Terms

First, identify the terms in the expression. We have two numbers: the positive number \(9.2\) and the negative number \(-11.4\).

Step 2: Determine Type of Addition

Since we're dealing with a positive number and a negative number, this isn't just an addition. We have to find the difference (or subtraction) because we're dealing with numbers on opposite sides of zero.

Step 3: Subtract the Smaller Number

Subtract the absolute value of the smaller number (9.2) from the absolute value of the larger number (11.4):\[ 11.4 - 9.2 = 2.2 \].

Step 4: Determine the Sign of the Result

The number with the larger absolute value was negative, so the result takes on this sign. Thus, the final result is negative: \(-2.2\).

Key Concepts

Absolute Value

When working with positive and negative numbers, understanding absolute value is crucial. The absolute value of a number is simply its distance from zero on the number line, without considering its direction. To put it simply, the absolute value doesn't care if a number is positive or negative; it only looks at how far away it is from zero.

• The absolute value of 9.2 is 9.2, because it is 9.2 units away from zero.
• Similarly, the absolute value of -11.4 is 11.4, as it is 11.4 units away from zero.

Understanding absolute value allows us to compare numbers based on their size without getting distracted by whether they are above or below zero. This concept is especially helpful when adding numbers with different signs, as it allows us to focus only on their magnitude.

Subtracting Numbers

In situations where you need to add a positive number to a negative number, you're actually finding the difference between the absolute values of these two numbers. This process is similar to subtraction. Subtract the smaller absolute value from the larger one to find the result.
Here's a step-by-step guide:

• Identify the absolute values of both numbers. For 9.2 and -11.4, these are 9.2 and 11.4, respectively.
• Subtract the smaller absolute value from the larger absolute value: \[11.4 - 9.2 = 2.2\]

This subtraction process simplifies the operation of adding a positive and a negative number into something more familiar and manageable. Once we have the result from this operation, we can move on to determining its sign.

Determining Sign of a Result

After performing the subtraction that results from adding a positive and a negative number, it is crucial to determine the sign of your answer. The rule here is quite straightforward: the resulting number will take the sign of the number with the larger absolute value.
In our example:

• We found the difference as 2.2.
• Since the larger absolute value (11.4) was originally negative (-11.4), the result also takes on the negative sign.
• Thus, the final result is \(-2.2\).

This method ensures you're applying the right sign to your answer, keeping your calculations consistent with the intuitive sense of "which number was stronger" in the operation. By following these steps, you can confidently and correctly find both the magnitude and the sign of your answer.