Problem 14 - Introductory and Intermediate Algebra for College Students 4th

Question

Find each sum without the use of a number line. $$-15+(-15)$$

Step-by-Step Solution

Answer

-30

Step 1: Identify the numbers

The two numbers are -15 and -15

Step 2: Apply the rule for adding negative numbers

When adding two negative numbers, you take the absolute value of both numbers, add those together, and then affix a negative sign to the result. So, |-15| + |-15| = 30, and then you affix a negative sign to get -30 as the result.

Key Concepts

Adding Negative Numbers

When dealing with negative numbers, many students find it tricky at first to understand how to add them, as it seems contrary to the usual operation of addition. However, it can be simplified with a basic rule.

To add two negative numbers, you simply need to find the absolute value of each. The absolute value of a number is that number without its sign.

• -15 becomes 15

Once you have the absolute values, add these numbers together just as you would if they were positive.

• In our example, 15 + 15 = 30

After obtaining the sum of the absolute values, don't forget to reapply a negative sign to your result. This is because both numbers you're adding were originally negative.

So, the result of (-15) + (-15) is -30. Think of it as combining two negative quantities, making the result more negative.

Absolute Value

The concept of absolute value plays a significant role in understanding and performing operations with integers, particularly when it comes to adding negative numbers.

The absolute value of a number is essentially its distance from zero on the number line, regardless of direction. It strips away the negative sign but maintains the magnitude.

• For example, the absolute value of -15 is 15 because it is 15 units away from zero.

This is usually denoted by vertical bars, such as \(-15\), meaning 'find the absolute value of -15.' Use absolute values to simplify calculations when dealing with operations involving negative integers.

• Instead of considering the sign immediately, find absolute values to simplify your additions.

Thus, understanding absolute values helps you navigate calculations by focusing on magnitude first, and reapplying signs afterwards.

Integer Operations

Integer operations include addition, subtraction, multiplication, and division of whole numbers and their negative counterparts. In algebra, understanding how these operations work with integers is foundational.

When working with addition of integers:

• If the numbers have the same sign, add their absolute values and keep the common sign. As illustrated in our case, \(-15 + (-15)\),we add the absolute values 15 + 15 to reach 30, then apply the negative sign.

If they have different signs, find the absolute value of each and subtract the smaller from the larger. The resulting sign is that of the original larger number.

• For instance, to solve 20 + (-15), subtract 15 from 20, yielding 5, and keep the sign of 20, resulting in 5.

Utilize these strategies to simplify and understand operations with integers, making algebraic problems more approachable.