Problem 67 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether each inequality is true or false. $$0 \geq-6$$

Step-by-Step Solution

Answer

The inequality \(0 \geq -6\) is true.

Step 1: Understanding Inequalities

Inequalities represent a relation between two expressions. Common symbols used are < (less than), > (greater than), \(\leq\) (less than or equal to), and \(\geq\) (greater than or equal to). The inequality symbol \(\geq\) represents 'greater than or equal to'. According to this, for the inequality \(a \geq b\), 'a' is greater than or equivalent to 'b'.

Step 2: Understanding number line

On a number line, positive numbers are always greater than negative numbers. 0 is greater than any negative number.

Step 3: Applying the inequality

In the case of the inequality \(0 \geq -6\), 0 is compared with -6. Since 0 is greater than any negative number including -6, the inequality \(0 \geq -6\) is true.

Key Concepts

Number Line

A number line is a visual representation of numbers on a straight line. It helps us understand the relative position of numbers and their order. Numbers increase as we move to the right and decrease as we move to the left.

In terms of positioning:

• Zero (\(0\)) is the central point, marking the division between positive and negative numbers.
• Positive numbers are located to the right of zero.
• Negative numbers are situated to the left of zero.

This visualization is particularly useful for comparing numbers, as you can easily see which number is greater based on their position on the line. By knowing that moving right means moving towards bigger numbers, we can conclude that zero is greater than any negative number just by where it stands compared to them.

Inequality Symbols

Inequality symbols are crucial because they help us express the relationship between numbers in terms of size. The common symbols include:

• \(<\): less than
• \(>\): greater than
• \(\leq\): less than or equal to
• \(\geq\): greater than or equal to

When we say \(a < b\), we mean 'a' is less than 'b'. Similarly, \(a \geq b\) means 'a' is greater than or equal to 'b'. This symbol tells us that the number on the left side is either bigger than the one on the right side or they are equal to each other. Understanding these symbols helps us communicate mathematical ideas succinctly.

Comparing Numbers

Comparing numbers is all about determining the size relation between them, like which one is bigger or smaller. This can always be checked using a number line or by knowledge of basic number properties.

When given two numbers:

• The number located more to the right on the number line is larger.
• The number with a bigger value without any negatives or additional symbols is greater.
• If one number is negative and the other is positive, the positive number is always greater.

For instance, when comparing \(0\) and \(-6\), zero is on the right side of negative six, and zero is also a whole number without negative, making it bigger. Hence, \(0 \geq -6\) is judged as a true statement, illustrating how these concepts of comparison translate into evaluating inequalities.