Problem 25 - Elementary Algebra

Question

Find the value of each of the following expressions. $$ 9(-2) $$

Step-by-Step Solution

Answer

Answer: -18

Step 1: Identify the mathematical operation

In this expression, we need to perform multiplication, specifically multiplying a positive number by a negative number.

Step 2: Apply multiplication rules for positive and negative numbers

When multiplying a positive number by a negative number, the result will be a negative number. The absolute value of the result is simply the product of the absolute values of the two numbers being multiplied. In this case, we have: $$ 9 \times (-2) $$

Step 3: Compute the result

Multiply the absolute values of the two numbers and add the negative sign to the result. Since the result will be negative, we have: $$ 9 \times (-2) = -(9 \times 2) = -18 $$ So the value of the expression is -18.

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers less than zero. They are commonly represented with a minus sign (-). For example, -1, -5, and -10 are all negative numbers. Negative numbers are used in various contexts, such as indicating temperatures below freezing, financial debts, or elevations below sea level. Unlike positive numbers, negative numbers decrease in value as they move further from zero. This means -10 is smaller than -2. When performing arithmetic operations with negative numbers, it's important to remember that:

• They behave differently in operations like addition and subtraction compared to their positive counterparts.
• When multiplied by positive numbers, the product is negative.
• When multiplied by another negative number, the product becomes positive.

Exploring Positive Numbers

Positive numbers are numbers greater than zero. They include all the natural numbers like 1, 2, 3, and so on, and are often used to count quantities or denoting measures above a reference point, like profits or heights above sea level. Positive numbers are typically written without any sign, but for clarity, they can be represented with a plus sign (+) in some contexts. When it comes to multiplication:

• Multiplying two positive numbers always results in a positive number.
• Conversely, as explained before, multiplying a positive number by a negative one always gives a negative result.

Understanding the nature of positive numbers helps in solving problems where you need to distinguish them clearly from negative numbers.

Introduction to Absolute Value

The absolute value of a number refers to its distance from zero on the number line, regardless of its direction (positive or negative). It's always a non-negative number. The absolute value is denoted by two vertical bars around the number, such as \(|a|\).For example:

• The absolute value of 5 is \(|5| = 5\).
• The absolute value of -3 is \(|-3| = 3\).

Absolute value is useful in simplifying expressions involving both positive and negative numbers, especially in multiplication and division. In multiplication, to find the result of positive and negative numbers, you first multiply their absolute values, which gives you the magnitude of the result, and then you determine the sign based on rules for multiplying negative and positive numbers.

Basic Arithmetic Operations and Multiplication Rules

Arithmetic operations are basic mathematical calculations that include addition, subtraction, multiplication, and division. Understanding these operations is crucial for solving various math problems. Focusing on multiplication, there are essential rules that govern the process, especially when dealing with positive and negative numbers:

• Multiplying two positive numbers yields a positive product.
• Multiplying two negative numbers also results in a positive product.
• However, if one number is positive and the other is negative, the product will always be negative.

In our exercise, multiplying 9 and -2 involves multiplying a positive number by a negative one, leading to a negative product. Remember, the absolute value helps us find the magnitude of the result, and the rule about the signs gives us the correct sign for the answer.