Problem 22 - Beginning and Intermediate Algebra

Question

Add. See Examples I through 7. $$ 53+(-37) $$

Step-by-Step Solution

Answer

16

Step 1: Identify the Numbers

The problem requires us to add two numbers: 53 and -37. Here, 53 is a positive integer and -37 is a negative integer.

Step 2: Understand the Operation

We need to add a positive and a negative integer. Adding a negative number is equivalent to subtracting its absolute value from the positive number.

Step 3: Subtract the Absolute Values

Subtract the absolute value of -37 from 53. This is equivalent to calculating 53 - 37.

Step 4: Perform the Subtraction

Calculate 53 - 37 by subtracting the smaller number from the larger one. 53 - 37 = 16

Step 5: Determine the Sign

Since the larger absolute value is from a positive number (53), the result takes the positive sign. Therefore, the result is positive 16.

Key Concepts

Positive and Negative Integers

In mathematics, integers are whole numbers that can be either positive or negative, including zero.

• Positive integers are numbers greater than zero, like 1, 2, and 53. They are located to the right side of zero on the number line.
• Negative integers are numbers less than zero, such as -1, -37, and -100. These numbers are found to the left of zero.
• Zero itself is neither negative nor positive; it is simply neutral.

Understanding the placement and behavior of positive and negative integers on the number line is crucial for performing operations like addition and subtraction. When you combine these numbers, one acts to move forward (positive) and the other backward (negative) along the line.

Subtraction

Subtraction is a basic arithmetic operation where you remove one value from another. It can be thought of as the opposite of addition. Subtraction can be visualized by thinking of it as counting backwards on a number line.

• When we subtract a smaller number from a larger one, the result is positive. For example, 10 - 4 = 6.
• If you subtract a larger number from a smaller one, the result is negative, such as 4 - 10 = -6.
• Importantly, subtracting a negative number can change the operation to addition. This is because subtracting a negative is like adding its positive counterpart. For example, 5 - (-3) becomes 5 + 3, resulting in 8.

Likewise, when encountering an expression like 53 + (-37), you can interpret it as 53 - 37, giving you a clear path to solve using simple subtraction.

Absolute Value

The absolute value of a number refers to its distance from zero on the number line, ignoring its direction or sign. It provides a way to measure magnitude without considering whether a number is positive or negative.

• The absolute value of a positive number is the number itself, such as \(|37| = 37\).
• For negative numbers, the absolute value is the positive version of the number, \(|-37| = 37\).
• The absolute value of zero remains zero because it is already neutral in terms of sign and distance.

Using absolute value simplifies operations with negative numbers, particularly in addition and subtraction. In our problem, adding negative integers becomes more straightforward when we subtract the absolute value of the negative number from the positive one. Thus, \(53 + (-37)\) is approached by calculating 53 minus \(|-37|\), which equals 16.