Problem 18 - Elementary Algebra

Question

Determine the product of -7 and -22 .

Step-by-Step Solution

Answer

The product is 154.

Step 1: Understand the Signs

When dealing with multiplication, it's important to understand the rule for signs. Multiplying two numbers with the same sign results in a positive product. So, multiplying two negative numbers will give us a positive result.

Step 2: Multiply the Absolute Values

Ignore the signs for a moment and multiply the absolute values of the two numbers. Calculate the product of 7 and 22. Thus, \(7 \times 22 = 154\).

Step 3: Apply the Sign Rule

Since both numbers were negative, the rule of multiplying two negatives gives us a positive result. Therefore, the product will be \(+154\).

Key Concepts

Negative Numbers

Negative numbers are numbers that are less than zero. They are typically represented with a minus sign (\(-\)). For example, \(-3\) and \(-15\) are negative numbers. Negative numbers are commonly found in various real-world situations, such as:

• Temperatures below freezing.
• Financial debts or losses.
• Altitudes below sea level.

Understanding negative numbers is crucial as they follow different rules compared to positive numbers in operations such as addition, subtraction, and, importantly for this exercise, multiplication. They indicate a quantity or value opposite in direction or value to a positive number.

Sign Rule in Multiplication

The sign rule in multiplication dictates how the signs of numbers affect the result of their product. When multiplying integers, the signs play an essential role in determining whether the result is positive or negative:

• A positive times a positive equals a positive.
• A negative times a negative also equals a positive.
• However, a positive times a negative equals a negative.

This happens because multiplying two negative numbers means you are taking the opposite of an opposite, which circles back to a positive value. In this exercise, multiplying \(-7\) and \(-22\), because both numbers are negative, follows the sign rule that dictates the product will end up being positive. Understanding this rule is vital as it can quickly help you determine the sign of your result.

Absolute Value

Absolute value refers to the distance of a number from zero on the number line, without considering direction. It is always a non-negative number. The absolute value is denoted by two vertical bars around the number: \(|-x|\) or \(|x|\). For example:

• The absolute value of \(-7\) is \(7\).
• The absolute value of \(3\) is \(3\).

In multiplication, focusing on the absolute values allows you to consider the size of the numbers without worrying about their signs first. It often simplifies calculations by treating positive and negative numbers as equivalent. In this exercise, you calculated the product of the absolute values \(7\) and \(22\) to get \(154\). The final step involved applying the sign rule to determine the correct sign of this product. Absolute values are useful for understanding and simplifying the problem at hand, thereby removing potential confusion with negative signs during multiplication.