Problem 59 - Elementary and Intermediate Algebra

Question

Classify each inequality as either true or false. $$ 0 \geq 8 $$

Step-by-Step Solution

Answer

False

Step 1: Understand the Symbols

The inequality symbol \(\textbackslash geq\) means 'greater than or equal to.' So, \(0 \textbackslash geq 8\) reads as '0 is greater than or equal to 8.'

Step 2: Evaluate the Inequality

To determine if the inequality \(0 \textbackslash geq 8\) is true, compare the quantities on both sides of the inequality. The left side is 0, and the right side is 8.

Step 3: Compare the Numbers

0 is not greater than 8. Also, 0 is not equal to 8.

Step 4: Conclusion

Since 0 is neither greater than nor equal to 8, we conclude that the inequality \(0 \textbackslash geq 8\) is false.

Key Concepts

Introducing Inequalities

Inequalities are mathematical expressions that show the relationship between two values. Inequalities use symbols to compare quantities. Common symbols include < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). These symbols help us understand how one number relates to another. Knowing these symbols is crucial when working with inequalities. For example, if we say '5 < 7', we mean 5 is less than 7. Knowing how to interpret these symbols is the first step in solving any inequality problem.
For complex problems, always start by identifying the symbols before moving on to compare values.

Understanding 'Greater Than or Equal To'

The symbol ≥ means 'greater than or equal to.' This symbol indicates that the value on the left side should either be greater than the value on the right side or equal to it. When you see ≥ in an equation or inequality, you need to consider two conditions:

• The value on the left is greater than the value on the right.
• The value on the left is equal to the value on the right.

For example, in the inequality 5 ≥ 3, 5 is greater than 3, so the inequality is true. In the inequality 7 ≥ 7, 7 is equal to 7, making the inequality true. Understanding this symbol ensures that you can accurately interpret and solve inequalities that contain it.

Evaluating Inequalities

Evaluating inequalities involves determining whether the inequality is true or false by comparing the values on both sides of the symbol. Here's a simple step-by-step guide to evaluate any inequality:

• Identify the values and the symbol: First, understand which numbers and inequality symbols are being used.
• Compare the numbers: Next, see if the numbers meet the criteria set by the inequality symbol. For example, check if one value is indeed larger than, less than, or equal to the other.
• Make a conclusion: Based on your comparison, decide whether the inequality is true or false.

Let's apply these steps to a concrete example: Inequality: 0 ≥ 8
1. The number on the left side is 0, and on the right side is 8. The symbol ≥ means we need to check if 0 is either greater than or equal to 8.
2. Compare 0 and 8. We know 0 is not greater than 8, and 0 is not equal to 8.
3. Thus, the inequality 0 ≥ 8 is false.
Mastering this process will enable you to confidently evaluate any inequality you encounter.