Problem 16 - Elementary and Intermediate Algebra

Question

Add. Do not use the number line except as a check. \(-8+(-12)\)

Step-by-Step Solution

Answer

-20

Step 1: Identify the Numbers

The first number is -8 and the second number is -12.

Step 2: Combine the Numbers

Since both numbers are negative, add their absolute values. The absolute value of -8 is 8 and the absolute value of -12 is 12. Adding 8 and 12 gives 20.

Step 3: Apply the Negative Sign

Since we added two negative numbers, the result will also be negative. Therefore, -8 + (-12) is -20.

Key Concepts

Absolute Value

When working with negative numbers, understanding absolute value is crucial. The absolute value of a number is its distance from zero on the number line, regardless of direction. So, whether the number is positive or negative, its absolute value is always positive.
For example, the absolute value of -8 is 8, and the absolute value of -12 is 12. This means -8 and -12 are each 8 and 12 units away from zero, respectively.
When adding negative numbers, we use their absolute values to simplify our calculations and combine them later with the appropriate sign.

Combining Like Terms

Combining like terms is a helpful algebraic process used to simplify expressions. In the context of adding negative numbers, we combine terms that are alike (both being negative in this case).
To illustrate, consider the problem \(-8 + (-12)\). In this problem:

• Both numbers are negative.
• We add their absolute values (\[ |−8| = 8 \] and \[ |−12| = 12 \]).
• The sum of 8 and 12 is 20.

Finally, since we were adding two negative numbers, the result is also negative, making the final result -20. Combining like terms helps us to efficiently handle operations involving similar components in mathematics.

Negative Integers

Negative integers are numbers less than zero and are often encountered in various mathematical problems. They are represented with a minus sign (−) in front of the number.
When adding negative integers, it is helpful to remember:

• Adding two negative integers will give a negative result. The absolute values of the numbers are summed up and the negative sign is kept.
• Negative integers represent a move left on the number line.

In, \(-8 + (-12)\), we are effectively moving left from zero by 8 units and then a further 12 units. This combined movement lands us at -20 on the number line.
Understanding these properties of negative integers helps us to navigate through and solve such problems more easily.