Problem 49 - Algebra and Trigonometry

Question

True or false. $$0 \geq-6$$

Step-by-Step Solution

Answer

The statement '0 ≥ -6' is true.

Step 1: Understand the inequality

To understand the inequality symbol ≥, you must remember that an inequality shows the relationship between two values that may not necessarily be equal. Symbol ≥ means 'greater than or equal to'. So, when 0 is compared to -6 using this symbol - it implies that 0 is either greater than -6 or equal to -6.

Step 2: Analyze the numbers

Now, consider 0 and -6 individually. Remember that 0 is greater than any negative number. Travelling to the right of the number line represents greater values. So, while -6 sits to the left on the number line, 0 is on the path to the right, making it larger than -6.

Step 3: Determine the truth of the statement

Since 0 is greater than -6, the statement '0 ≥ -6' holds true.

Key Concepts

Inequality Symbol

Inequalities are mathematical expressions that show the relationship between two values when they are not equal. The inequality symbol is essential in comparing these values. There are several types of inequality symbols, such as:

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

Each symbol helps us understand how values relate to one another beyond just equality.
For example, the symbol \( \geq \) suggests not only that one number could be greater, but also that it might be exactly equal. This is important in mathematics as it accounts for multiple possibilities within a range, allowing for more comprehensive expressions of relationships between numbers.

Greater Than or Equal To

The 'greater than or equal to' symbol, expressed as \( \geq \), provides a powerful way to convey two distinct possibilities in one concise expression.
This symbol is used when one quantity is either larger than another or exactly the same. Suppose you have a statement like "0 \( \geq \) -6", it implies that:

• 0 is greater than -6 because it represents larger values on a number line.
• 0 could also be considered equal in contexts where these numbers match, though here there isn't practical equality as they are distinct values.

Understanding this concept helps in evaluating and solving problems correctly, and is fundamental in mathematical reasoning.

Number Line

A number line is a visual representation of numbers laid out in a straight line, which is extremely helpful in understanding relationships between numbers.
Here's how it works:

• Numbers increase in value as you move from left to right.
• Negative numbers occur on the left, zero is in the middle, and positive numbers are on the right.

In the example of "0 \( \geq \) -6", we start at -6 on the number line and move rightward to 0. The movement to the right indicates increasing value, so 0 being to the right of -6 confirms that 0 is indeed greater.
The number line is an intuitive tool for seeing inequalities.

Negative Numbers

Negative numbers are values less than zero and are represented by a minus sign. They are essential in various mathematical contexts and contrast with positive numbers and zero.
Important features of negative numbers include:

• They appear to the left of zero on the number line.
• The further left they are, the smaller their value. For example, -6 is less than -1.
• They are often used to represent a deficit or loss.

In the case of our inequality "0 \( \geq \) -6", understanding negative numbers helps us realize that any positive number, zero included, is always greater than negative numbers.
This concept is critical in comparing numbers and solving inequalities accurately.