Problem 118 - Intermediate Algebra

Question

Translate into mathematical symbols: ten less than twice \(x\)

Step-by-Step Solution

Answer

The expression is \(2x - 10\).

Step 1: Understanding 'Twice x'

The term 'twice' means two times a number. If the number is represented by the variable \(x\), then 'twice \(x\)' is represented as \(2x\).

Step 2: Understanding 'Ten Less Than'

The phrase 'ten less than' indicates that ten is subtracted from some quantity. For example, if you have a quantity \(Q\), \'ten less than \(Q\)' is represented as \(Q - 10\).

Step 3: Constructing the Expression

Combine the information from Steps 1 and 2. We want 'ten less than twice \(x\)'. First write 'twice \(x\)' as \(2x\), then subtract 10 from \(2x\), resulting in the expression \(2x - 10\).

Key Concepts

Translation of Phrases

Understanding how to translate phrases into mathematical expressions is essential in algebra. It's like learning a new language, where words turn into symbols and operations. In our example, the phrase "ten less than twice a number" needs to be translated step by step into math language. First, identify the terms of the phrase. Here, "twice" and "ten less than" guide the mathematical operations involved. By recognizing these key terms, you can convert the entire phrase into a clear and concise mathematical expression.

Variables

In algebra, variables are symbols used to represent unknown or changeable values. They are typically represented by letters like \(x\), \(y\), or \(z\). These symbols allow for generalization in mathematical expressions and equations.

• **Common Variables**: Letters such as \(x\), \(y\), \(z\) are often used.
• **Purpose**: To solve problems where actual numbers are unknown.
• **Flexibility**: Variables can take on different values, making them versatile.

In the exercise, \(x\) is the variable representing the number whose relation to ten times itself we aim to express.

Arithmetic Operations

Arithmetic operations are the foundation of forming and solving mathematical expressions. They include addition, subtraction, multiplication, and division. In solving problems, recognizing which operation to use is key.

• **Addition**: Signs like \(+\), implying sum or togetherness.
• **Subtraction**: Depicted by \(-\), indicating removal or reduction.
• **Multiplication**: Shown by \(\times\) or just placement, for instance, \(2x\).
• **Division**: Symbolized by \(/\), suggesting separation into parts.

In our problem, the operations used involve multiplication ('twice') and subtraction ('ten less than'), illustrating how these keywords or phrases guide the operations.

Algebraic Expressions

Algebraic expressions are the combination of variables, numbers, and operations that represent a particular quantity or relationship. They are not equations, but rather ways to express ideas mathematically.

• **Components**: Involve constants, variables, and arithmetic operations.
• **Purpose**: Describe and signify the relationships between numbers and variables.
• **Interpretation**: Requires understanding the structure and terminology.

The exercise results in the expression \(2x - 10\), clearly demonstrating the shift from verbal descriptions to a mathematical formulation, capturing the intended meaning behind the phrase.