Problem 15 - Elementary and Intermediate Algebra

Question

Add. Do not use the number line except as a check. \(-6+(-5)\)

Step-by-Step Solution

Answer

The sum of -6 and -5 is -11.

Step 1: Identify the Numbers

Identify the two numbers that need to be added. Here, the numbers are -6 and -5.

Step 2: Understand Addition with Negative Numbers

Adding two negative numbers together makes the sum more negative. In essence, you are adding the absolute values of the numbers and then appending a negative sign to the result.

Step 3: Add the Absolute Values

Find the absolute values of -6 and -5. The absolute value of -6 is 6, and the absolute value of -5 is 5. Add these absolute values together: 6 + 5 = 11.

Step 4: Apply the Negative Sign

Since both original numbers were negative, the result will also be negative. Therefore, the result will be -11.

Key Concepts

negative numbers

Negative numbers are numbers less than zero. They are represented with a minus or negative sign (-). For example, -5 is a negative number. When you look at the number line, negative numbers are found to the left of zero. The more to the left you go, the smaller the value. For example, -10 is less than -3.
Negative numbers are often used to show debts, temperatures below freezing, and elevations below sea level.
In arithmetic, negative numbers can be added, subtracted, multiplied, and divided just like positive numbers, but the rules for dealing with the signs need to be followed carefully.

absolute value

The absolute value of a number is its distance from zero on the number line, without considering the direction. It is always a positive number or zero. The absolute value is represented using vertical bars. For example, the absolute value of -6 is written as \(|-6|\) and equals 6.
In general, the absolute value of a number 'a' is denoted as \(|a|\). If the number inside the bars is negative, you simply remove the minus sign to find the absolute value. For instance, \(|-7|\) is 7, while \(|3|\) is 3.
When dealing with problems involving negative numbers, understanding how to find and use the absolute value is crucial.

integer addition

Integer addition involves combining whole numbers, which include positive numbers, negative numbers, and zero. The rules for adding integers are straightforward but must be followed carefully:

• When adding two positive integers, just add their values. For example, \(6 + 3 = 9\).
• When adding two negative integers, add their absolute values and then apply a negative sign to the result. For instance, in the exercise \(-6 + (-5)\), you add \(|-6| = 6\) and \(|-5| = 5\), resulting in 11. Since both integers are negative, the sum is \(-11\).
• When adding a positive integer and a negative integer, subtract the smaller absolute value from the larger absolute value. Keep the sign of the number with the larger absolute value. For example, \(-7 + 4\) would be calculated as \(|(-7)| - |4| = 7 - 4 = 3\), and since -7 has the larger absolute value, the result is negative, so the answer is \(-3\).

Mastering these rules will help you to add integers accurately.