Problem 70 - Introductory Algebra for College Students

Question

Determine whether each inequality is true or false. $$-14 \geq 8$$

Step-by-Step Solution

Answer

The inequality '-14 ≥ 8' is false.

Step 1: Understanding the inequality symbol

The inequality symbol '≥' is used when the value on the left is either greater than or the same as the value on the right.

Step 2: Compare the two values

In this case, -14 is on the left side and 8 is on the right side of the inequality. -14, being a negative number, is smaller than 8, which is a positive number.

Step 3: Verifying the inequality

Since -14 is not greater than or equal to 8, the given statement, '-14 ≥ 8', is false.

Key Concepts

Understanding Inequality Symbols

In algebra, understanding inequality symbols is fundamental for comparing values and solving problems. These symbols serve as the shorthand for expressing the relationship between two values. The most common inequality symbols include:

• \textless{} which means 'less than',
• \textgreater{} for 'greater than',
• \textless{}= for 'less than or equal to',
• \textgreater{}= denoting 'greater than or equal to'.

When we observe an inequality like \( -14 \geq 8 \), we're encountering the \(\geq\) symbol, indicating that the value on the left should be greater than or equal to the one on the right for the statement to be true.
Exploring the meanings of these symbols is crucial for interpreting and solving inequality problems in algebra, since they guide us to the comparative nature of the values in question.

Comparing Values in Inequalities

To properly compare values in inequalities, we must interpret the numbers in relation to each other based on the inequality symbol used. A positive number is always greater than a negative number. Addressing the exercise \( -14 \geq 8 \), we compare the two numbers -14 and 8. Here, -14 represents a quantity less than zero, while 8 is a positive number, signifying a value more than zero and therefore greater than -14. In the comparison process, it's essential to recognize the fundamental number line where values increase from left to right and any number to the left is smaller than a number to the right. By keeping these principles in mind, students can accurately determine the truth value of inequalities.

Verifying Inequalities

Verifying an inequality involves checking if the statement holds true based on the values and the symbol used. After understanding the symbol and comparing the values, one must decide if the inequality accurately reflects the relationship. In our case, \( -14 \geq 8 \) suggests that -14 should be greater than or equal to 8. However, since -14 is a negative number and 8 is a positive number, the mathematical fact that any positive number is greater than any negative number leads to a clear-cut conclusion. The statement is indeed false because -14 is not greater than or equal to 8. Verifying inequalities often requires a practical understanding of how numbers relate to each other on a basic level, which in turn reinforces one's ability to solve more complex algebraic problems.