Problem 103 - Introductory Algebra for College Students

Question

Determine whether this inequality is true or false: \(19 \geq-18 .\) (Section 1.3, Example 7)

Step-by-Step Solution

Answer

The inequality \(19 \geq -18\) is true.

Step 1: Compare the Numbers

19 is a positive number and -18 is a negative number. By definition, any positive number is greater than any negative number. So, without any calculations, we can assert that 19 is indeed greater than -18.

Step 2: Evaluate the Inequality Statement

Using the comparison result from Step 1, we can definitively say that the inequality \(19 \geq -18\) is a true statement, because 19 (a positive number) is greater than -18 (a negative number).

Key Concepts

Positive and Negative Numbers

When dealing with mathematics, especially inequalities, understanding positive and negative numbers is crucial. A number line helps illustrate where these numbers fall relative to one another. Generally, numbers to the right of zero on the number line are positive, and those to the left are negative.

Here's a quick breakdown:

• Positive numbers are greater than zero and appear to the right on a number line.
• Negative numbers are less than zero and are represented to the left of zero on a number line.
• Zero itself is neutral—neither positive nor negative.

It's important to remember that any positive number, no matter how small, is larger than any negative number, no matter how large in absolute value. This foundational concept makes it easy to assert that positive numbers, like 19 in our exercise, are always greater than negative numbers, such as -18.

Algebraic Inequalities

Algebraic inequalities are expressions that involve variables or numbers connected by inequality signs like greater than (">") or less than ("<"). Understanding these inequalities helps you determine the relationship between different values.

In our particular example, the inequality is presented as \(19 \geq -18\). This inequality uses the greater than or equal to sign ("\( \geq \)"). It indicates that 19 is either greater than or equal to -18.

Here’s a refresher on how to interpret these symbols:

• \( > \) means "greater than"
• \( < \) means "less than"
• \( \geq \) indicates "greater than or equal to"
• \( \leq \) signifies "less than or equal to"

By comparing the numbers involved in the inequality, it becomes clear that 19 is greater than -18, making the statement \(19 \geq -18\) true.

Comparing Numbers

Comparing numbers is an integral part of solving inequalities. You'll often use this skill to determine the order or magnitude of different values, essential for both simple arithmetic and complex algebraic equations.

To effectively compare numbers:

• Use a number line to visualize their relative positions.
• Remember that positive numbers are always greater than negative numbers.
• Focus on the magnitude, keeping the sign in mind.

When comparing numbers like 19 and -18, it's intuitive to recognize that the presence of a negative sign will always make a number less than a positive one. This basic understanding allows you to quickly verify inequalities such as \(19 \geq -18\) without extensive calculations. In practice, comparing numbers is a valuable skill across various math problems and real-world scenarios.