Problem 107 - Elementary and Intermediate Algebra

Question

Translate each phrase to mathematical symbols. Let \(x\) represent the unknown number. a. 4 times a number increased by 2 b. 4 times a number, increased by 2

Step-by-Step Solution

Answer

Both expressions translate to \(4x + 2\).

Step 1: Define the unknown number

Let's assume the unknown number is represented by \(x\). This will be our variable throughout the translation process.

Step 2: Identify mathematical operations for part (a)

The phrase is '4 times a number increased by 2'. This translates to the expression \(4x + 2\). It implies you first multiply the number \(x\) by 4 and then add 2.

Step 3: Identify mathematical operations for part (b)

The phrase '4 times a number, increased by 2' suggests a slight difference in grouping. First, \(4x\) is computed and then you increase the total by 2. However, structurally it expresses the same sequence as part (a): \(4x + 2\).

Key Concepts

Mathematical Symbols

Mathematical symbols are the notations and characters used to represent mathematical concepts and operations. In algebra, these symbols form the building blocks for expressing relationships and operations on numbers. For instance, in the expressions derived from the exercise, the symbol "4x" involves both numbers and letters.
"4" is a numerical coefficient, indicating that the number it is paired with is multiplied by 4. The symbol "+" is an operator showing addition, and it connects terms within an expression, such as adding 2 in our scenario.
These symbols help simplify complex words and phrases into concise mathematical language. When translating phrases to symbols, it’s crucial to carefully consider the operations implied by the textual description to accurately reproduce the expression.

Variables in Algebra

In algebra, variables are symbols, often letters, used to represent unknown numbers or values that can change. They serve as placeholders, allowing us to express general laws and relationships applicable to various numerical instances. In the given exercise, the variable is represented by "\(x\)".
This letter is used to stand in for the unknown number in both parts of the original problem. When working with variables, it's important to maintain consistent naming conventions throughout an exercise or equation. This consistency helps avoid confusion and ensures clear communication of mathematical ideas.
Variables can take on any numerical value, and through manipulation, solving algebraic expressions allows you to find these unknown values, bridging the gap between abstract math and concrete numbers.

Mathematical Operations in Algebra

Mathematical operations in algebra are the processes that involve manipulation and calculation on numbers or variables; they are the engine rooms of algebraic expressions. The primary operations include addition, subtraction, multiplication, and division.
In the phrases given in the exercise, we encounter several of these operations:

• Multiplication is indicated by the term "times" and is represented in the expression by "4x", which means 4 times the variable \(x\).
• Addition occurs with the inclusion of "increased by 2", represented as "+ 2" in the expression.

These operations follow the rules and hierarchy established in mathematics, often governed by the order of operations. Understanding and correctly applying these in sequence are essential for translating word phrases into accurate mathematical symbols. For example, recognizing that multiplication takes precedence over addition is vital for interpreting expressions correctly.