Problem 61 - Elementary Algebra

Question

Use algebraic notation to write the statement "six times a number is less than or equal to eleven."

Step-by-Step Solution

Answer

Question: Translate the statement "six times a number is less than or equal to eleven" into algebraic notation. Answer: 6x ≤ 11

Step 1: Define the variable

Let x be the unknown number mentioned in the statement.

Step 2: Write the inequality

Now, we need to express the statement "six times a number is less than or equal to eleven" as an inequality. We can write this as: 6x ≤ 11. So, the algebraic notation for the given statement is 6x ≤ 11.

Key Concepts

Variable Definition

When working with algebraic notation, defining variables is the first step. In the context of mathematical problems, a "variable" is a symbol, often a letter like \(x\), used to represent an unknown value or number that we are trying to find or describe. This allows for generalization and manipulation of equations.

In our example exercise, we defined \(x\) as the unknown number referenced in the statement. This process of choosing a letter to stand for the unknown provides the basis for writing mathematical expressions and solving problems. It is important to choose a variable that is easy to use and understand, as it will serve as our placeholder in subsequent mathematical expressions.

• Choose a simple letter, commonly \(x\), but any letter can be used.
• Clearly state what the variable represents to avoid confusion later.

Variables are key in forming equations and inequalities, serving as the link between the problem statement and its mathematical notation.

Inequality

Inequalities in mathematics are expressions where two values are compared, often to show that one is lesser or greater than the other. They have specific symbols to denote these relationships:

• \(<\) for "less than",
• \(>\) for "greater than",
• \(\leq\) for "less than or equal to",
• \(\geq\) for "greater than or equal to".

In our exercise, the statement "six times a number is less than or equal to eleven" was translated into the inequality \(6x \leq 11\). This shows a clear relationship between the product of six and our unknown number \(x\) and the number eleven.

Understanding inequalities is crucial, as they are used frequently in both educational and real-world scenarios. They help us establish boundaries and limits and are pivotal in fields ranging from finance to physics. The use of the correct inequality sign is vital for accurately communicating the relationship described.

Mathematical Expressions

Mathematical expressions are combinations of numbers, operations, and sometimes variables, used to represent a value or relationship. They can be as simple as a single number or involve complex operations and variables.

In the context of our exercise, "six times a number" translates into the expression \(6x\). This indicates the operation (multiplication) applied to the variable (x) and a constant (six). Mathematical expressions can be manipulated to solve equations or translate real-world scenarios into solvable formats.

Here are some key points:

• Expressions can include numbers, operators (+, -, *, /), and variables.
• They represent quantities and relationships.
• Expressions are the building blocks for equations and inequalities.

Having a strong grasp of how to construct and interpret mathematical expressions is fundamental for solving problems in algebra and beyond. By translating words into math, we create a universal language for solving problems.