Problem 53 - Elementary Algebra

Question

True or false. \(1 \neq 1\)

Step-by-Step Solution

Answer

False, because 1 is equal to 1.

Step 1: Understand the Symbols

The expression \(1 eq 1\) uses the inequality symbol \( eq \), which means "not equal to." The expression is stating that 1 is not equal to 1.

Step 2: Evaluate the Expression

To evaluate the expression \(1 eq 1\), compare the two numbers on both sides of the inequality symbol: you have the number 1 on both sides, which are exactly the same.

Step 3: Determine the Truth Value

Since the number 1 is indeed equal to 1, the expression \(1 eq 1\) is false. In mathematical terms, when something is equal but written with a 'not equal' operator, the statement is false.

Key Concepts

Inequality Symbols

Inequality symbols are fundamental tools in mathematics that help us compare two values or expressions. They are used to express a relationship that is not strictly equal. The most common inequality symbols are:

• \( eq \) (not equal to): Indicates that two values are not equal.
• \( < \) (less than): Represents that the value on the left is smaller than the one on the right.
• \( > \) (greater than): Shows that the left side value is larger than the right side.
• \( \leq \) (less than or equal to): Means the left side value is smaller than or the same as the right.
• \( \geq \) (greater than or equal to): The left value is larger than or the same as the right.

Each symbol helps express conditions that we encounter in real-life problems or theoretical calculations. Knowing the meaning of each can help us understand and create mathematical expressions that describe the world accurately.

Truth Value Evaluation

Truth value evaluation is a key part of deciding whether a mathematical statement is true or false. In our example, we want to decide if the statement \(1 eq 1\) is true.

• Start by looking at the values or expressions on each side of the inequality symbol. Here, both sides of \(1 eq 1\) are the number 1.
• Since both sides are indeed equal, the inequality statement is false.

Evaluating truth values involves comparing the actual values and determining if the relationship stated is indeed the case. If our inequality symbol represents something not true (like in our \(1 eq 1\) case), then the statement holds a false truth value.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and symbols placed together to make a meaningful statement. They might involve:

• Numbers: These include constants or coefficients like 1, 2, 5, or any numeric value.
• Variables: Often represented by symbols like \(x\), \(y\), or \(z\) which stand in for unknown values.
• Operators: Such as "+", "-", "\(*\)", or "/" which are used to perform arithmetic operations.
• Symbols: Like equality \((=)\) and inequality \((eq, <, >)\) symbols, to represent relationships.

In mathematics, expressions serve as the building blocks of equations and inequalities, letting us state problems and relationships in a concise and clear form. We should always look carefully at each part of an expression to correctly interpret the overall meaning and solve or simplify it efficiently.