Problem 23 - Beginning and Intermediate Algebra

Question

Add. See Examples I through 7. $$ -18+49 $$

Step-by-Step Solution

Answer

The result is 31.

Step 1: Identify the Numbers and their Signs

Examine the given expression \[-18 + 49\]to identify the two numbers, -18 and +49, and note their signs: -18 is negative, and 49 is positive.

Step 2: Perform the Arithmetic Operation

To add a negative number and a positive number, you subtract the absolute values of these numbers, ignoring their signs initially. Here, subtract the absolute value of 18 from the absolute value of 49:\[49 - 18 = 31\]

Step 3: Determine the Sign of the Result

Since the larger absolute value (49) originally had a positive sign, the result of the operation will also be positive. Thus, the result is +31.

Key Concepts

Absolute Value

Understanding absolute value is important when adding integers because it gives the number's size without considering its sign. Simply put, the absolute value is always a non-negative number. For example, the absolute value of

• \(-18\) is \(18\)
• \(+49\) is \(49\)

Both of these absolute values tell us how far the numbers are from zero on a number line, regardless of direction.
When you're adding numbers with different signs, like \[-18 + 49\], focusing on absolute values helps us decide what arithmetic operation to perform. In this case, since one number is negative and the other is positive, we can subtract the smaller absolute value from the larger one.

Arithmetic Operations

In the context of adding integers like \[-18 + 49\], arithmetic operations are the steps needed to combine the numbers correctly. Here, you initially ignore the signs and perform subtraction of absolute values because one number is negative and the other is positive.
Remember this simple rule:

• When adding numbers with the same sign, add their absolute values and keep the sign.
• When adding numbers with different signs, subtract the smaller absolute value from the larger one and keep the sign of the larger absolute value.

For the example, you would calculate \[49 - 18 = 31\], then determine the sign of the result based on the larger absolute value's original sign.

Positive and Negative Numbers

Positive and negative numbers follow simple yet crucial rules that dictate how they behave in arithmetic operations, especially addition.

Positive numbers are usually larger than zero (e.g., 1, 2, 3, ...), and negative numbers are smaller than zero (e.g., -1, -2, -3, ...). When you encounter an operation like \[-18 + 49\], you see one negative and one positive number.
This situation leads us back to the rule: when adding numbers with different signs, subtract their absolute values. Since \(+49\) has a larger absolute value than \(-18\), the result, in absolute terms, will be positive.

• Keep the sign of the number with the larger absolute value.
• In this case, \[18\] has a smaller absolute value compared to \[49\], so the overall sum is positive, resulting in\[+31\].

Fully understanding these rules allows for quick and accurate calculations in a wide variety of math problems.