Problem 34 - Elementary and Intermediate Algebra

Question

Add. Do not use the number line except as a check. \(-20+(-6)\)

Step-by-Step Solution

Answer

-26

Step 1: Identify the Operation

Understand that you are asked to add two negative numbers: \(-20 + (-6)\).

Step 2: Add the Absolute Values

Take the absolute values of the numbers, which are 20 and 6. Add these absolute values together: 20 + 6 = 26.

Step 3: Apply the Sign

Since both original numbers are negative, the result will also be negative. So, the sum of \(-20 + (-6)\) will be -26.

Step 4: Double-Check on the Number Line

Although the problem advises not to primarily use the number line, it's good to check the result by starting at -20 and moving 6 units to the left, ending at -26.

Key Concepts

absolute value

Absolute value refers to the distance of a number from zero on the number line. Importantly, this distance is always a positive number or zero.

In mathematical terms, the absolute value of a number is denoted by two vertical lines. For instance, the absolute value of -20 is written as \(|-20| = 20\). Similarly, the absolute value of 6 is \(|6| = 6\).

Understanding absolute values is crucial when dealing with negative numbers. When adding \(-20 + (-6)\), you first ignore their signs. You focus on the absolute values - 20 and 6. Then, add these absolute values to get 26. Using the absolute value helps simplify the problem before applying the final sign based on the original numbers.

negative integers

Negative integers are numbers less than zero. They have a minus sign in front of them, such as -1, -20, or -6.

When adding negative numbers, it's like combining debts. For example, if you owe someone 20 dollars and then borrow another 6 dollars, your total debt increases. Mathematically, this is shown as \(-20 + (-6) = -26\).

The key steps are:

• Ignore the signs and add the absolute values.
• Since both numbers are negative, the result also must be negative.

So, when you find that 20 + 6 equals 26, you reapply the negative sign to get the answer: -26.

number line

A number line is a visual tool that helps illustrate the concept of numbers, their sequences, and their operations.

It is a straight line with points that correspond to numbers. Positive numbers are on the right of zero, and negative numbers are on the left.

Even though the exercise suggests not relying on the number line, it is helpful for checking our work. For instance, starting at -20 and moving 6 units to the left reflects the addition \(-20 + (-6)\). You end at -26.

This visual movement confirms the solution \(-20 + (-6) = -26\). Thus, the number line assists in understanding and validating our calculations visually.