Problem 65 - Introductory Algebra for College Students

Question

Determine whether each inequality is true or false. $$-9 \geq-9$$

Step-by-Step Solution

Answer

The inequality -9 ≥ -9 is true.

Step 1: Understanding the inequality symbol

The symbol '≥' means 'greater than or equal to'. This means the value on the left side of the symbol can either be greater than or equal to the value on the right side for the inequality to hold true.

Step 2: Compare the values

We have -9 to the left side of '≥' and -9 to the right side. These are equal values, which means -9 is equal to -9.

Step 3: Evaluate the inequality

Since the left and right values, which are both -9, are equal and the inequality symbol '≥' allows for equality, the given inequality, -9 ≥ -9, is true.

Key Concepts

Greater Than or Equal To

The concept of 'greater than or equal to', represented by the symbol \( \geq \), is fundamental to understanding inequalities. When we see this symbol, it indicates that the value on the left is either larger than or exactly equal to the value on the right. This is important in mathematics because it allows us to express a range of possible solutions.

For instance, if we encounter \( a \geq b \), it means:

• \( a \) is greater than \( b \)
• or \( a \) is equal to \( b \)

Understanding this dual condition is crucial as it incorporates both strict inequality and the possibility of equality.
In our exercise, \( -9 \geq -9 \), since \( -9 \) is equal to \( -9 \), the inequality holds true.

Comparison of Values

A key step in working with inequalities is comparing values on both sides of the inequality sign. This determines whether the inequality holds true or not. Here, we compare \( -9 \) on the left side of \( \geq \) with \( -9 \) on the right.

Comparison involves examining numeric or variable expressions to see if one is larger, smaller, or equal to the other. In this specific case, both sides are identical, meaning they are equal. Understanding this allows you to correctly interpret the inequality symbol. Comparisons can be made easily when numbers are straightforward, but they might involve simplifications or rearrangements when variables come into play.
Think of comparing values as a critical step to uncovering the relationship between numbers in an inequality.

Evaluating Inequalities

Evaluating inequalities involves checking whether the mathematical statement is true or false. This process builds on understanding the inequality symbol and comparing values. Our exercise \( -9 \geq -9 \) is a good example.

Here’s how to evaluate:

• Identify the left and right values of the inequality.
• Compare them to see if they satisfy the condition set by the inequality symbol.
• If they do, the statement is true; otherwise, it's false.

This involves substituting variables with numbers where necessary, performing calculations, and logically analyzing the results.
By breaking it down into simple steps, evaluating inequalities becomes a more straightforward task, paving the way for handling more complex problems.