Problem 58 - Elementary Algebra

Question

True or false. $$ 3>3 $$

Step-by-Step Solution

Answer

False.

Step 1: Understanding the Inequality

The given inequality is \(3 > 3\). Inequalities compare two values and use symbols: \(>\) for greater than, \(<\) for less than, \(\geq\) and \(\leq\) for greater than or equal to and less than or equal to, respectively. Here, we are checking if 3 is greater than 3.

Step 2: Evaluating the Expression

Since both numbers on either side of the inequality are equal (both are 3), the statement \(3 > 3\) suggests that 3 is greater than itself. In mathematics, a number is never greater than itself; rather, it is equal to itself.

Step 3: Concluding the Truth of the Statement

Since the statement \(3 > 3\) suggests one number is greater than the other but they are equal, the statement is not true. Therefore, the given inequality is false.

Key Concepts

The Greater Than Symbol

In mathematics, symbols play a key role in conveying complex ideas succinctly. The greater than symbol, denoted as \(>\), is one such essential symbol used in inequalities. It indicates that the number on the left side is larger than the number on the right.
Let's break it down:

• Greater than: Signified by \(>\), it means the value on the left is more than the value on the right.
• Examples: In \(5 > 3\), 5 is greater than 3, confirming the statement is true.
• Usage: Typically checks if a value exceeds another, helping to order numbers or expressions.

Remember, the direction of the sign is important. Pointing towards the lesser value ensures clarity and accuracy.

Understanding Mathematical Statements

A mathematical statement is a sentence with a truth value, either true or false. It conveys a precise idea or relation using numbers and symbols.
Here’s how it works:

• Structure: Includes variables, constants, and symbols to express a logical idea.
• Examples: "\(x + 2 = 5\)" is a mathematical statement indicating that when \(x\) is 3, the equality holds true.
• Purpose: These statements make mathematics clearer and structured, allowing us to solve problems effectively.

When tackling these statements, assess their components carefully to determine their validity.

Evaluation of Expressions

Evaluating expressions involves determining the value of a mathematical phrase based on its components. This process is crucial in solving equations and testing the truth of inequalities.
Let's walk through it:

• Components: Consist of numbers, variables, and operations that together form a coherent mathematical phrase.
• Evaluation: Means to calculate or simplify the expression to find its numeric value or check its truthfulness.
• Example: In the expression \(3 > 3\), evaluation reveals both sides are equal, thus proving the inequality false.

By being diligent in evaluating expressions, mathematical reasoning becomes robust and reliable.