Problem 43 - Prealgebra and Introductory Algebra

Question

Add. $$-25+(-31)+24+19$$

Step-by-Step Solution

Answer

-13

Step 1: Identify the positive and negative integers

In the given exercise, -25 and -31 are negative integers, while 24 and 19 are positive integers.

Step 2: Adding negative integers

Add the negative integers together. When we add two negative integers, we add their absolute values and put a negative sign in front of the sum. So, \(-25 + (-31)= - (25+31) = -56 \) .

Step 3: Adding positive integers

Add the positive integers together, which is as simple as regular addition. So, \(24 + 19 = 43\).

Step 4: Final Sum

Add the sum of negative integers to the sum of positive integers, which is actually a subtraction (-56 from 43). So, \(-56 + 43 = -13\). So, \(-25 + (-31) + 24 + 19 = -13\).

Key Concepts

Understanding Positive and Negative Integers

In mathematics, integers are whole numbers that can be either positive or negative. Positive integers are numbers greater than zero, such as 1, 2, 3, etc. Negative integers are numbers less than zero, like -1, -2, -3, and so on. Understanding the difference between these two types of integers is crucial for solving addition problems that involve both.
When working with these numbers, we notice that positive numbers add to a total value, whereas negative numbers subtract from a total.
Recognizing whether an integer is positive or negative helps us know whether to add or subtract when combining values.

The Concept of Absolute Value

Absolute value refers to the distance of a number from zero on the number line, without considering direction.
For example, both 5 and -5 have the same absolute value, which is 5. This is because both numbers are 5 units away from zero on the number line.
When adding negative numbers, the absolute value helps determine how far and in which direction they move.

• The absolute value of any positive number is the number itself.
• The absolute value of any negative number is the number without its negative sign.

Using absolute value is especially helpful when adding negative numbers together.

Addition of Negative Numbers

Adding negative numbers might seem tricky at first, but it's straightforward once you understand the process.
When you add two negative numbers, you are essentially adding their absolute values together and then applying a negative sign to the result.
For example, to add -25 and -31, you calculate the absolute values, 25 and 31, sum them to get 56, and then make the result negative: -56. This process works because adding a negative number means moving further away from zero in the negative direction.

• Negative + Negative = More Negative

So, -25 + (-31) equals -56 because you move further in the negative direction.

Addition of Positive Numbers

Adding positive numbers is perhaps one of the easiest operations in math. It is simply straightforward addition, adding the numbers to find the total.
For instance, when you have the numbers 24 and 19, you just add them: 24 + 19 = 43.
This total represents how much you gain or accumulate when dealing with quantities or counts.

• Positive + Positive = Larger Positive

Positive integer addition doesn't require worrying about signs; it's all about increasing the total value by simply bringing numbers together.