Problem 9 - Intermediate Algebra

Question

Perform the following operations with real numbers. $$ -8-14 $$

Step-by-Step Solution

Answer

-22

Step 1: Understand the Problem

The problem involves subtracting two real numbers. We have an expression: \(-8 - 14\). Both numbers are negative, so we treat this as subtraction in the context of real numbers.

Step 2: Perform the Subtraction

To subtract \(-14\), we change the sign of 14 and add it to -8. In terms of addition of negative numbers, this expression transforms to \(-8 + (-14)\).

Step 3: Add the Numbers

Add the numbers by keeping the sign and adding the absolute values: \(-8 + (-14) = - (8 + 14) = -22\). Thus, the result of the operation is \(-22\).

Key Concepts

Addition and Subtraction of Real Numbers

Real numbers include all rational and irrational numbers. These numbers can be positive, negative, or zero. When performing addition and subtraction with real numbers, it is important to keep track of their signs and absolute values. This process involves understanding the rules for combining positive and negative numbers.

• When adding two positive numbers, the result is always positive.
• When adding a positive number and a negative number, you subtract the smaller absolute value from the larger one and keep the sign of the number with the larger absolute value.
• When adding two negative numbers, the result is negative. You simply add their absolute values and apply a negative sign to the result.
• Subtraction of a number is the same as adding its negative, i.e., to subtract a number, add its opposite.

In our exercise, \[-8 - 14\] is transformed using the rule of subtracting by adding the negative, \[-8 - 14 = -8 + (-14)\].
Then, you add the absolute values of 8 and 14 to find the result \[-(8+14) = -22\].

Negative Numbers

Negative numbers are numbers less than zero. They are used to represent a deficiency or a reversal in mathematical calculations. Some essential points to understand about negative numbers include:

• Negative numbers are the opposite of positive numbers.
• On a number line, they appear to the left of zero.
• When adding two negative numbers, their absolute values are summed, and the result is also negative.
• The subtraction involving negative numbers can often be thought of as adding the opposite sign.
• For instance, subtracting -14 becomes adding +14 when performed with a negative number like -8.

In the problem from our example, both numbers involved are negative: \[-8\] and \[-14\].
This means we need to think of the operation as adding two negative amounts, leading to a further negative total, \[-22\].

Absolute Value

The absolute value of a number refers to its distance from zero on the number line, regardless of direction. It is always a non-negative number.
Here are key points about absolute value:

• The absolute value of a positive number is the number itself.
• The absolute value of a negative number is the positive version of the number. For example, the absolute value of \(-8\) is \(8\).
• The absolute value of zero is \(0\).
• Absolute values are vital in calculations involving addition or subtraction of negative numbers, as they help us understand the "magnitude" without concerning the sign.

In our step-by-step solution, \(|-8| = 8\) and \(|-14| = 14\) are used to calculate the sum \(8 + 14 = 22\).
The understanding of absolute values allows us to correctly apply the sign of the result, leading to \(-22\), as both numbers being added are negative.