Problem 49 - College Algebra Essentials

Question

Determine whether each statement in Exercises 43–50 is true or false. $$0 \geq-6$$

Step-by-Step Solution

Answer

The statement '0 \(\geq\) -6' is true.

Step 1: Understand the Inequality

The inequality in question pits 0 against -6 with a \(\geq\) sign in between them. This means that we are checking whether 0 is greater than or equal to -6.

Step 2: Compare the Numbers

0 is greater than any negative number since numbers increase in value from left to right on the number line. Therefore, zero is greater than -6.

Step 3: Evaluate the Statement

Given that 0 is indeed greater than -6, we can confidently say that the statement '0 \(\geq\) -6' is true.

Key Concepts

Understanding the Number Line

When it comes to inequalities, the number line is a powerful tool. A number line is a straight line with numbers placed at equal intervals along its length. It extends infinitely in both directions. These numbers include all integers, fractions, and decimals.

• Positive numbers are to the right of zero.
• Negative numbers are to the left of zero.
• Zero is the center, separating positive and negative numbers.

The further right you move on the line, the greater the value of the number. Conversely, the further left you move, the lower the value. This visual representation makes it easy to compare numbers and understand which is greater or lesser. For example, any positive number is greater than zero. Similarly, zero is greater than any negative number.

Interpreting 'Greater Than or Equal To'

In inequalities, ">=" is a crucial symbol known as "greater than or equal to." It indicates that the value on the left is either greater than, or precisely equal to, the value on the right. This dual condition is what sets it apart from a simple "greater than" (>).In the inequality expression "0 \( \geq \) -6," we are tasked with determining if zero is greater than or equal to negative six. Given our knowledge of the number line, zero is, in fact, greater than any negative number. Therefore, this statement holds true.In more general terms:

• If the inequality symbol is satisfied by equality (like 0 = 0), that means the statement is true.
• If the expression without the equal part holds true (like 0 > -6), the statement is also true.

Working with Negative Numbers

Negative numbers can sometimes be tricky, but understanding them is essential. A negative number is any number less than zero. They are located to the left of zero on the number line. Here are a few characteristics:

• Negative numbers decrease in value the further left you move.
• A negative number is always less than zero.
• The more negative a number is, the smaller its value.

For instance, -3 is greater than -6 because it is closer to zero on the number line. When comparing zero and any negative number, zero is always greater. This understanding is crucial when evaluating inequalities that involve negative numbers.

Composing Mathematical Statements

Mathematical statements often communicate relationships between numbers using symbols. These symbols form the language of mathematics, letting us express ideas concisely.Key symbols include:

• "=" for equality.
• ">" for greater than.
• ">=" for greater than or equal to.

To make sense of a statement like "0 \( \geq \) -6," we follow a logical approach:1. Identify the numbers involved (0 and -6).2. Understand the symbol used (\( \geq \)).3. Compare the numbers using their positions on the number line.This process helps determine whether a statement is true or false. In our case, 0 is indeed greater than -6, confirming the truth of the mathematical statement."