Problem 69 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether each inequality is true or false. $$-17 \geq 6$$

Step-by-Step Solution

Answer

The inequality is false. It is not correct to say that -17 is greater than or equal to 6.

Step 1: Interpret the inequality

The inequality -17 ≥ 6 reads as 'negative seventeen is greater than or equal to six.' Therefore, the task is to determine if negative seventeen is indeed greater than or equal to six.

Step 2: Compare the numbers

In numerical order from lesser to bigger, negative numbers always come before positive numbers. Therefore, -17 is not greater than nor equal to 6.

Key Concepts

Understanding Inequalities

Inequalities are mathematical expressions used to compare two values or quantities. When working with inequalities, the purpose is to determine the relationship between these values. Inequality symbols include the following:

• \( > \): Greater than
• \( \, < \): Less than
• \( \geq \): Greater than or equal to
• \( \leq \): Less than or equal to
• \( eq \): Not equal to

Mathematicians and students, use inequalities to express conditions that involve numerical relationships. For instance, the expression \(-17 \geq 6\) uses the "greater than or equal to" symbol \(\geq\) to show a comparison between -17 and 6.

When analyzing inequalities, your task is to evaluate the truth of these statements. Did you know understanding inequalities is crucial in fields like economics and data science? It's because comparing values efficiently allows for deeper insights into trends and predictions.

The Role of Negative Numbers

Negative numbers are numbers less than zero. They hold a critical position in mathematics. Unlike positive numbers, which increase to the right on a number line, negative numbers decrease to the left. This characteristic impacts how we interpret inequalities involving negative numbers.

When comparing any negative number to a positive one, it is essential to remember:

• Any negative number is always less than any positive number. For example, -5 is less than 3.
• The further left a negative number is from zero, the less its value. For example, -20 is less than -5.

This makes it clear why \(-17\) is not greater than, nor equal to, 6. In mathematics, visualizing a number line can be helpful. Negative numbers' placement is consistently to the left of positive numbers, making them automatically smaller.

Grasping Numerical Order

Numerical order is the sequence in which numbers sit on a number line. It's important for comparing numbers and understanding their relative positions. Numbers arranged from smallest to largest show their natural order, starting with negative numbers, moving through zero, and extending to positive numbers.

Here's what you need to know:

• Negative numbers always appear first, as they are less than positive numbers.
• The order follows: largest negative numbers come first, then smaller negative numbers, zero, and finally positive numbers.
• This specific order impacts how we read and analyze inequalities.

So, for the inequality \(-17 \geq 6\), remember that -17's position is far left compared to 6. Hence, it is confirmed as false since it doesn't satisfy the condition of being greater than or equal to 6, reflecting its actual placement in the numerical sequence.