Problem 47 - College Algebra Essentials

Question

Determine whether each statement in Exercises 43–50 is true or false. $$-\pi \geq-\pi$$

Step-by-Step Solution

Answer

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

Step 1: Understand the operator

The operator ≥ denotes that the value on the left is greater than or equal to the value on the right. This is a non-strict inequality, meaning the values can also be equal.

Step 2: Symbol substitution

Here, we are given \(-\pi \geq-\pi\). As the value of both sides is \(-\pi\), it implies the left-hand side is equal to right-hand side, which satisfies the condition of the operator.

Step 3: Result

Therefore, \(-\pi \geq-\pi\) is True as \(-\pi\) is indeed equal to \(-\pi\).

Key Concepts

Greater than or equal to

The symbol \( \geq \) represents a **greater than or equal to** comparison. It is a way of expressing that one quantity is larger than or equal to another. For example, if we say \( a \geq b \), it means \( a \) can be larger than \( b \) or equal to \( b \). Here are more details to consider:

• The "greater than" aspect implies the value on the left is larger than the right.
• The "equal to" part of \( \geq \) indicates the values can be the same.

This kind of expression is widely used to show that two values can either be identical or one can be higher. This concept is important in algebra, calculus, and more, allowing us to compare numbers and derive conclusions based on these relationships.

Non-strict inequality

When we use terms like **non-strict inequality**, we are talking about inequalities that allow equality as a possibility. Non-strict inequalities include symbols such as \( \leq \) (less than or equal to) and \( \geq \) (greater than or equal to). This distinguishes them from strict inequalities like \( < \) and \( > \), where equality is not an option.Non-strict inequalities:

• Allow equality (\( a \geq b \) or \( a \leq b \))
• Are flexible in comparisons, useful in situations where either quantity can change dynamically
• Provide a broader understanding of two quantities' relationship

By using non-strict inequalities, we can easily adapt to different scenarios in mathematics where the values being compared might sometimes be the same.

Equality

**Equality** is a fundamental concept in mathematics, represented by the symbol \( = \). When two expressions or numbers are equal, it means they have the same value or represent the same quantity. In our exercise, this was demonstrated by considering whether \( -\pi \) is equal to \( -\pi \). It is!Key characteristics of equality include:

• Both sides of an equation have identical values.
• It reflects balance and symmetry between two sides.
• Identifying equality allows you to solve equations by ensuring balance.

Equality is a powerful tool, not just for solving equations, but for understanding relationships and ensuring that mathematical proofs remain consistent and valid.