Problem 16 - Elementary and Intermediate Algebra

Question

Multiply. $$ 9 \cdot(-5) $$

Step-by-Step Solution

Answer

-45

Step 1 - Understand the Problem

Identify the two numbers being multiplied. In this case, the numbers are 9 and -5.

Step 2 - Multiplication Rule with Negative Numbers

Recall the rule for multiplying a positive number by a negative number: the result is always negative.

Step 3 - Perform the Multiplication

Multiply the absolute values of the numbers: \(9 \times 5 = 45\). Since one of the numbers is negative, the result is negative.

Step 4 - Write the Result

Combine the result with the negative sign: \(-45\).

Key Concepts

Negative Numbers

Negative numbers are numbers that are less than zero. They are usually written with a minus sign (-) in front of them. For example, -1, -2, and -5 are negative numbers. Negative numbers are used to represent a variety of real-world situations, such as temperatures below freezing, debts, and elevations below sea level.
When dealing with negative numbers, it's important to remember that they behave differently from positive numbers in certain calculations. For instance, when you add two negative numbers, the result is another negative number. Similarly, when you subtract a larger negative number from a smaller negative number, the result is positive.

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always a non-negative number. For example, the absolute value of -5 is written as \(|-5|\) and the result is 5.
Absolute value essentially turns negative numbers into positive numbers while leaving positive numbers unchanged. This concept is useful in various mathematical operations, including multiplication and solving equations. It helps simplify the process by allowing you to focus on the magnitude of the numbers without worrying about their signs at first.

Multiplication Rules

When multiplying integers, the rules for the signs are crucial. Here are the basic multiplication rules for integers:

• Positive \( \times \) Positive = Positive
• Positive \( \times \) Negative = Negative
• Negative \( \times \) Negative = Positive

In the given exercise, we multiply 9 (a positive number) by -5 (a negative number). According to the rules, the result is negative. To perform the multiplication, first find the absolute values: \(|9| = 9\) and \(|-5| = 5\). Then multiply these absolute values: \(9 \times 5 = 45\). Since the original problem involves a positive and a negative number, the final result is \-45\.