Problem 12 - Elementary and Intermediate Algebra

Question

Multiply. $$ -3 \cdot 7 $$

Step-by-Step Solution

Answer

-21

Step 1: Understand the signs

Review the multiplication of signs. A negative number multiplied by a positive number gives a negative result.

Step 2: Multiply the absolute values

Forget about the signs for a moment and multiply the absolute values. Here, the absolute values are 3 and 7.

Step 3: Perform the multiplication

Multiply 3 by 7 to get 21.

Step 4: Apply the sign rule

Since the original problem involves a negative number (-3) and a positive number (7), the result will be negative. Therefore, the answer is -21.

Key Concepts

Negative and Positive Numbers

In mathematics, numbers can be either negative or positive. A positive number is one that is greater than zero and is usually written without a sign or with a '+' sign. Negative numbers are less than zero and are written with a '-' sign.
Let's imagine numbers as steps on a number line:

• Positive numbers move you to the right of zero.
• Negative numbers move you to the left of zero.

The concept is important because it affects the result of arithmetic operations, especially multiplication.

Absolute Values

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always a positive number. For instance:

• The absolute value of -3 is 3, written as \(|-3| = 3\).
• The absolute value of 7 is 7, written as \(|7| = 7\).

When multiplying, you often first consider the absolute values, ignoring the signs. This makes the calculations easier before you apply any sign rules.

Multiplication Rules

Understanding the rules for multiplying integers is crucial:

• Positive times Positive equals Positive: \( (+) \times (+) = + \)
• Negative times Negative equals Positive: \( (-) \times (-) = + \)
• Positive times Negative equals Negative: \( (+) \times (-) = - \)
• Negative times Positive equals Negative: \( (-) \times (+) = - \)

So, for the exercise -3 \( \cdot \) 7:

1. Consider the signs: A negative times a positive gives a negative result.
2. Multiply the absolute values: \(|-3| = 3\) and \(|7| = 7\).
3. Perform the multiplication: \(3 \cdot 7 = 21\).
4. Apply the correct sign: Since we have a negative and a positive number, the result is -21.

This ensures you get the correct answer.