Problem 81 - Algebra A Combined Function

Question

Fill in the box with \(<,>, \leq,\) or \(\geq\). See the Concept Check in this section. Since \(3<5,\) then 3(-4)\(\square 5(-4)\)

Step-by-Step Solution

Answer

3(-4) > 5(-4)

Step 1: Understand the Problem

We need to compare the expressions \(3(-4)\) and \(5(-4)\) and fill in the box with the appropriate inequality sign \(<, >, \leq,\) or \(\geq\).

Step 2: Calculate Each Expression

Calculate \(3(-4) = -12\) and \(5(-4) = -20\).

Step 3: Compare the Results

Now, compare the two results. We have \(-12\) and \(-20\). Since \(-12\) is greater than \(-20\), the correct inequality is \(>\).

Step 4: Fill in the Box

Since \(3(-4) = -12\) and \(5(-4) = -20\), we fill in the box with \(3(-4) > 5(-4)\).

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers less than zero. They are usually represented with a minus sign (-) placed before a digit. Negative numbers appear commonly in various mathematical operations and can sometimes be a bit tricky to work with, especially when involved in more complex expressions.

When visualizing negative numbers, you can think of them as points on a number line extending to the left from zero. These numbers have properties and behaviors that differ from their positive counterparts.

• Addition: Adding negative numbers results in a decrease of the value. For example, the sum of -3 and -2 is -5.
• Subtraction: Subtracting a negative number is equivalent to adding its positive version. For instance, -5 - (-3) becomes -5 + 3, which equals -2.
• Comparison: On a number line, a negative number is smaller than any positive number. Among negative numbers, the one closer to zero is greater. For example, -2 is greater than -5.

A solid understanding of negative numbers is fundamental when solving inequalities. This concept is essential when multiplying or comparing expressions that involve negative values.

Mastering Multiplication with Negative Numbers

When multiplying numbers, and particularly negative numbers, understanding the signs is crucial. When two numbers are multiplied, the resulting sign depends on the signs of what you started with.

• Positive multiplied by Positive: The result is positive (e.g., 3 × 4 = 12).
• Positive multiplied by Negative: The result is negative (e.g., 3 × -4 = -12).
• Negative multiplied by Negative: The result is positive (e.g., -3 × -4 = 12).

In the given exercise, we multiply positive numbers by the same negative number (-4). This means both products will be negative:

• 3 × -4 results in -12.
• 5 × -4 results in -20.

This rule of signs is pivotal. Understanding multiplication helps inferring the solution when comparing negative products, as one encountering such questions in solving inequalities.

Comparing Expressions Involving Negatives

When faced with algebraic expressions that include negative numbers, the main goal is to determine which expression is larger or smaller.

The exercise you encountered involved expressions resulting in negative numbers - namely, -12 and -20. To compare expressions effectively:

• First, calculate or simplify each expression individually, bearing in mind the rules for negative numbers and multiplication.
• Align them on a number line in your mind or on paper. Recall that the farther left on the number line, the smaller the value.
• Compare the results. Here, -12 is greater than -20, as -12 is closer to zero compared to -20.

This understanding enables us to place the correct inequality symbol between two expressions. In this scenario, the box is appropriately filled with a ">" between 3(-4) and 5(-4), as -12 exceeds -20 in value.