Problem 55 - Elementary Algebra

Question

True or false. $$ 4 \leq 4 $$

Step-by-Step Solution

Answer

True.

Step 1: Understanding the Symbols

First, identify the symbols in the expression. The symbol "\(\leq\)" is a comparison operator meaning "less than or equal to". It compares two values and checks if the first value is either less than or equal to the second value.

Step 2: Analyzing the Expression

In the given expression, we have \(4 \leq 4\). This implies checking if 4 is less than or equal to 4. Since 4 is equal to 4, this part of the comparison is true.

Step 3: Conclusion

Since 4 is equal to 4, and the statement checks for either 'less than' or 'equal to', the statement \(4 \leq 4\) is indeed true.

Key Concepts

Comparison Operators

Comparison operators are essential in mathematics for comparing the values of numbers and determining relationships between them. They help us state how two values relate to each other. The most commonly used comparison operators are:

• \(< >\): Greater than
• \(< \lt \): Less than
• \(\geq\): Greater than or equal to
• \(\leq\): Less than or equal to
• \(=\): Equal to
• \(eq\): Not equal to

In the context of the exercise, the operator "\(\leq\)" symbolizes 'less than or equal to'. When we say \(4 \leq 4\), we are checking if 4 is either less than or exactly equal to 4. These operators are straightforward but crucial for building and evaluating mathematical expressions.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operators that collectively represent a specific value or condition. In our example, the expression \(4 \leq 4\) features a straightforward relationship between two numbers and a comparison operator.
Key components of a mathematical expression include:

• Numbers: These are constants like 4 in our example.
• Operators: These include arithmetic (+, -, *, /) and comparison operators (like \(\leq\)).
• Variables: In more complex expressions, symbols like \(x\) or \(y\) represent unknowns or values that can change.

A mathematical expression can be as simple as a single number or as complex as an equation involving multiple operations. Simplifying and understanding an expression involves identifying its elements and how they relate.

True or False Statements

A true or false statement in mathematics evaluates a condition to determine its validity. These statements are often used to check relationships or solve equations. A true statement means the condition holds, while a false statement indicates it does not.
In the expression \(4 \leq 4\):

• We evaluate if 4 is less than 4 (which is false), or
• We evaluate if 4 is equal to 4 (which is true).

Since the 'equal to' part of the condition is true, the entire statement is true. Understanding true or false statements involve recognizing the components and testing the logical conditions specified. This process helps in verifying the solutions to mathematical problems efficiently.