Problem 47 - College Algebra

Question

Determine whether statement is true or false. \(-\pi \geq-\pi\)

Step-by-Step Solution

Answer

The statement \(-\pi \geq-\pi\) is true.

Step 1: Interpret the Inequality

The inequality \(-\pi \geq-\pi\) is comparing two equal numbers. It asks if negative pi is greater than or equal to negative pi. In any number system, a number is always equal to itself.

Step 2: Evaluate the Inequality

Given that any number is equal to itself, \(-\pi\) is indeed equal to \(-\pi\). Therefore, the statement is true because the inequality allows for \(-\pi\) to be equal to \(-\pi\).

Key Concepts

Real Numbers

Real numbers are an essential part of mathematics that include all the numbers you can think of, such as integers, fractions, and an infinite variety of numbers with decimal points. They can be either positive or negative and can be represented on a number line.

• Integers: Whole numbers such as -3, 0, and 5.
• Rational numbers: Numbers that can be expressed as a fraction, like 1/2 or 0.75.
• Irrational numbers: Numbers that cannot be written as a simple fraction, such as \(\u03C0\) or \(\u221A2\).

Understanding real numbers helps us explain quantities that are both irrational and not immediately obvious, like \(\u03C0\), which represents the ratio of a circle's circumference to its diameter.
In this exercise, \(-rac{\pi}{1}\) is considered a real number because it can be placed on the number line, representing a specific value.

Inequality Evaluation

Inequality evaluation involves determining whether one number is greater than, less than, or equal to another. The various symbols used in inequalities include:

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

In the exercise \(-\pi \geq-\pi\), we have the symbol \(\geq\), which indicates that the first number must be greater than or equal to the second.
Since \(-\pi\) is equal to \(-\pi\), the inequality is satisfied, making the statement true. This demonstrates that inequations consider equality when using the \(\geq\) or \(\leq\) symbols.

Number Properties

Understanding number properties allows us to comprehend mathematical statements better, especially when addressing variables and constants. Key properties include:

• Commutative Property: Numbers can be added or multiplied in any order.
• Associative Property: The way numbers are grouped in addition or multiplication does not change their sum or product.
• Reflexive Property: Any number is always equal to itself, i.e., \(a = a\).
• Transitive Property: If \(a = b\) and \(b = c\), then \(a = c\).

The reflexive property plays a vital role in confirming the truth of \(-\pi \geq-\pi\). Since a number is always equal to itself, the inequality allows for equivalence, ensuring it holds true. Recognizing these properties simplifies the evaluation of mathematical expressions and inequalities.