Problem 43 - Beginning and Intermediate Algebra

Question

Add See Examples \(\ell\) through 7 . $$ -21+(-16)+(-22) $$

Step-by-Step Solution

Answer

The sum is -59.

Step 1: Identify the Integers

The problem gives us three integers to add together: -21, -16, and -22. Pay attention to the signs of each number.

Step 2: Understand Addition of Negative Numbers

When adding negative numbers, the result is more negative. We are adding three negative numbers, which means the final result will also be negative.

Step 3: Add the First Two Numbers

Start by adding the first two numbers: \(-21 + (-16) = -37\). Think of this as moving 21 steps to the left on a number line, and then another 16 steps to the left.

Step 4: Add the Third Number

Now add the result from Step 3 to the third number: \(-37 + (-22) = -59\). This means moving another 22 steps to the left from -37.

Key Concepts

Negative Numbers

Negative numbers can often seem puzzling at first, but they are quite straightforward if you think of them as numbers just like positive ones, only on the opposite side of zero. On a number line, negative numbers are located to the left of zero. It’s like having a long stick on which zero is the middle point, with positive numbers stretching to the right and negative numbers to the left. When dealing with negative numbers, remember:

• Negative numbers represent values less than zero.
• The more negative the number, the further it is from zero on the number line.
• Negative numbers are used for many real-world situations, such as temperatures below freezing, bank account overdrafts, or descents below sea level.

Understanding the nature of negative numbers is crucial for adding them correctly, as you will move consistently in one direction along the number line when working with them.

Addition of Integers

When adding integers, especially when they include negative numbers, you can think of this operation as moving along a number line. Each integer represents a step:

• Positive integers mean steps to the right.
• Negative integers mean steps to the left.

For the exercise you have:

• First, add \(-21 + (-16)\). Both numbers are negative, so move 21 steps and then another 16 steps to the left. This results in \(-37\).
• Next, use the result \(-37\) and add \(-22\) by moving 22 more steps left, resulting in \(-59\).

Adding negative numbers increases the distance you move away from zero, thus making the number more negative. Always focus on the direction you are moving on the number line to determine the final result.

Number Lines

A number line is a very helpful visual tool for understanding the addition and subtraction of integers. It is a straight line where each point corresponds to a number. Here’s why it helps:

• It clearly shows the relative position of positive and negative numbers.
• Allows you to visualize the process of moving left or right when adding and subtracting integers.

When working with addition of integers, especially with negatives:

• Start at zero or your initial number.
• For \(-21 + (-16)\), move 21 steps left from zero and then 16 more steps to reach \(-37\).
• For \(-37 + (-22)\), continue moving 22 steps left from \(-37\) to arrive at \(-59\).

Using a number line simplifies complex calculations by allowing you to see the movement and relationship between numbers. It's an excellent method to grasp the concept of integer addition with negative numbers.