Problem 129 - Algebra A Combined Function

Question

State whether each statement is true or false. The product of three negative integers is negative.

Step-by-Step Solution

Answer

The statement is true.

Step 1: Understand the Question

The statement asks us to determine if the product of three negative integers results in a negative value. We should recall the rules about multiplying negative numbers to evaluate this.

Step 2: Review Multiplication Rules for Negatives

When multiplying two negative numbers, the product is positive, because a negative times a negative results in a positive. However, when you multiply one more negative number (making it three altogether) with this positive product, the result becomes negative again.

Step 3: Consider an Example

Let's take three negative integers, for example, \(-2\), \(-3\), and \(-4\). Calculate the product step by step:1. Multiply the first two integers: \(-2 \times -3 = 6\) (positive)2. Multiply this result by the third integer: \(6 \times -4 = -24\) (negative)Thus, the product of these three negative integers is negative.

Step 4: Conclude Based on Rules and Example

Based on the general rule and our example, we can see that multiplying three negative integers always results in a negative product. This matches the situation described in the statement.

Key Concepts

Multiplication Rules

When it comes to multiplying numbers, understanding the relationship between negative and positive numbers is essential. There are specific rules that help us determine the result when multiplying these numbers. Here are the fundamental rules of multiplication involving negative numbers:

• Multiplying two positive numbers always yields a positive result.
• Multiplying two negative numbers results in a positive product. This might seem a bit counterintuitive, but think of it as a double negation.
• Multiplying a positive number by a negative number, or vice versa, results in a negative product.

These rules serve as a foundation to predict the product of various combinations of positive and negative numbers. Applying them correctly allows you to determine the sign of the resulting product without even needing to calculate it fully at first.

Negative Integers

Negative integers are whole numbers that are less than zero. They are located to the left of zero on the number line. These numbers are often preceded by a minus sign (-). Examples of negative integers include -1, -2, -3, and so on. Understanding these is vital for arithmetic operations. In multiplication, negative integers play a unique role because they affect the sign of the product. Consider these examples:

• For the pair of negative integers -5 and -3, the product is a positive 15 because multiplying two negatives results in a positive.
• Adding another negative integer, such as -2, into the multiplication turns the product negative: (15) × -2 = -30.

The rules for combining these numbers are straightforward once you understand the multiplication rules. Being adept with negative integers will help in effortlessly solving multiplication problems.

Product of Integers

The term 'product' refers to the result of multiplying two or more numbers together. When dealing with integers, which include positive whole numbers, negative whole numbers, and zero, their signs play a significant role in determining the product. Let's explore the concept of integer products further.When combining multiple integers, the resulting product's sign is positive if there is an even number of negative factors and negative if there is an odd number. Here are a few points to remember:

• Three negative integers multiplied together will result in a negative product. For example, consider the integers -2, -3, and -4. Their product is calculated as: \[ (-2) \times (-3) = 6 \quad (positive) \] \[ 6 \times (-4) = -24 \quad (negative) \]
• No matter the number of positive integers involved, if they are mixed in with an odd number of negative integers, the result will stay negative.

By keeping track of the number of negative factors involved, one can easily determine the sign of the product of integers involved.