Problem 23 - Beginning and Intermediate Algebra

Question

Multiply. $$ 4(-7) $$

Step-by-Step Solution

Answer

The product is -28.

Step 1: Identify the Elements

In this multiplication problem, we have a positive number 4 and a negative number -7 that need to be multiplied.

Step 2: Multiply the Absolute Values

First, ignore the signs of the numbers and multiply the absolute values. The absolute value of 4 is 4 and the absolute value of 7 is 7. So, calculate \( 4 \times 7 \).

Step 3: Calculate the Product

Multiply the absolute values calculated in the previous step: \( 4 \times 7 = 28 \).

Step 4: Apply the Sign Rule

Since we are multiplying a positive number by a negative number, the product will be negative. Thus, the product of 4 and -7 is -28.

Key Concepts

absolute value

Understanding the concept of absolute value is crucial in mathematics, especially when dealing with the multiplication of integers. The absolute value of a number is its distance from zero on the number line, without considering its direction (whether positive or negative). It’s like how far you are from home, not caring if you went north or south. For instance, the absolute value of both \(4\) and \(-4\) is \(4\), because each is four units away from zero.

When you multiply integers, initially, you might need to ignore their signs. For example, consider the exercise \(4(-7)\). You start by taking the absolute value of each number. Here, the absolute value of \(4\) is \(4\), and the absolute value of \(-7\) is \(7\). You then multiply these values just like you would in regular multiplication: \(4 \times 7 = 28\).

This method helps simplify the process by focusing first on the magnitude (size) of the numbers, before worrying about the sign.

sign rule in multiplication

The sign rule in multiplication is an essential principle that helps determine the sign of the product. When multiplying two integers, the product's sign is determined by the signs of the numbers involved. There are specific rules to follow, which are summarized as follows:

• If you multiply two positive numbers, the product is positive.
• If you multiply two negative numbers, the product is also positive because two negatives make a positive.
• If you multiply a positive number by a negative number, the product is negative.
• Conversely, multiplying a negative number by a positive number also yields a negative product.

In the exercise \(4(-7)\), you multiply a positive number (\(4\)) by a negative number (\(-7\)). According to the rule, since the signs are different, the product is negative. Thus, you find the result to be \(-28\). Understanding this rule simplifies problems involving integer multiplication.

algebra

In algebra, understanding how to operate with numbers using their fundamental properties is vital. Algebra extends basic arithmetic by introducing variables (often represented by letters) and allows us to create mathematical expressions and equations.

Multiplication of integers is a common operation in algebra that requires careful consideration of both absolute values and sign rules. Often, algebraic expressions include mixed signs, and it becomes important to apply both the absolute value and sign rule concepts properly to simplify or solve them.

For example, if given \(x = 4\) and \(y = -7\), when you solve the expression \(xy\), you apply the absolute values and sign rules we’ve discussed. You calculate \(4 \times 7 = 28\) first and then apply the appropriate sign based on their rules, resulting in \(-28\).

Being able to manipulate expressions in this way by breaking them into understandable steps is part of what makes algebra a powerful tool for solving a wide range of mathematical problems.