Problem 11 - Beginning and Intermediate Algebra

Question

Multiply. $$ 2(-9) $$

Step-by-Step Solution

Answer

-18

Step 1: Understand the Task

The exercise asks to multiply two numbers: 2 and -9. Negative numbers behave differently from positive numbers, especially regarding their signs.

Step 2: Multiply the Magnitudes

Ignore the signs first, and multiply the absolute values of the numbers. Multiply 2 by 9. \[ 2 imes 9 = 18 \]

Step 3: Determine the Sign of the Product

Since you are multiplying a positive number by a negative number, the product will be negative according to the rule: a positive times a negative equals a negative. Therefore, the result of 2 times -9 will carry a negative sign. So, the result is -18.

Key Concepts

Negative Numbers

Negative numbers are numbers that are less than zero. Often, they are represented with a minus sign (-) in the front. These numbers are used to express values below a baseline, like temperatures below freezing, or financial debts.
It's important to distinguish between positive and negative numbers because they behave differently in mathematical operations.

• When you add a negative number, it's like subtracting its absolute value.
• Subtracting a negative number is like adding its absolute value.
• Multiplying or dividing two negative numbers results in a positive number.

In the exercise, the number -9 is negative, illustrating the distinct feature that multiplying it with a positive number requires specific attention to the signs.

Rules of Multiplication

The rules of multiplication dictate how to handle the signs of the numbers being multiplied. These rules are important for determining the result's sign:

• A positive number times a positive number is positive.
• A negative number times a positive number is negative.
• A positive number times a negative number is also negative.
• A negative number times a negative number results in a positive number.

In our example, we multiply a positive number 2 by a negative number -9. According to the rules, a positive number times a negative number yields a negative result.
Therefore, the product of 2 and -9 is -18. Understanding these rules helps simplify the process and avoid mistakes in calculations.

Absolute Values

Absolute value refers to the distance of a number from zero on the number line, without considering its direction (or sign). It's denoted by vertical bars around the number, like this: \(|-9|\).

• For positive numbers and zero, the absolute value is the number itself.
• For negative numbers, the absolute value is the positive version of the number.

In the exercise, to find the product of 2 and -9, we first ignore the signs and multiply their absolute values: \( |2| \times |-9| = 2 \times 9 = 18 \). After this calculation, we then apply the rules of multiplication to decide on the sign of the result.
Using absolute values in this initial step ensures the magnitude of the result is correct before considering the sign.