Problem 2 - Prealgebra

Question

Write each of the following English phrases in symbols using the variable \(x\). The difference of \(x\) and 2

Step-by-Step Solution

Answer

The expression is \(x - 2\).

Step 1: Identify the Phrase Components

The phrase 'The difference of \(x\) and 2' has two main components: 'difference,' which means subtraction, and the numbers involved—\(x\) and 2.

Step 2: Translate 'Difference' into Symbols

The term 'difference' in mathematics implies subtraction. We will express 'difference of \(x\) and 2' as \(x - 2\).

Step 3: Write the Expression

Combine the components identified to write the expression. Using the translation from Step 2, 'The difference of \(x\) and 2' can be written symbolically as \(x - 2\).

Key Concepts

Variable Expressions

In prealgebra, a **variable expression** is a mathematical phrase that contains numbers, operations, and variables instead of values alone. Unlike equations, these expressions do not have an "equals" sign. The oxygen for these expressions are the variables, which act like placeholders. They are often denoted by letters such as

• \( x \)
• \( y \)
• \( z \)

These variables represent unknown quantities that we can substitute with different numbers depending on the situation. As seen in our exercise, the expression "the difference of \( x \) and 2" translates into the variable expression \( x - 2 \). Here, \( x \) is the variable. Such expressions are powerful tools because they allow us to generalize rules and solve for unknowns in a flexible manner.

Mathematical Symbols

**Mathematical symbols** are the shorthand used in mathematics to convey complex ideas quickly and accurately. They are universal, meaning that anyone familiar with math can understand them regardless of their native language. The essential symbols we'll often encounter include:

• \( + \) for addition
• \( - \) for subtraction
• \( \times \) or \( \cdot \) for multiplication
• \( \div \) for division

In our exercise, the symbol \(-\) is a key player as it represents the difference or subtraction, denoted in the phrase "the difference of \( x \) and 2". This tulip of a symbol helps us create the expression \( x - 2 \) efficiently. Using symbols, we can transition smoothly from worded problems to mathematical expressions and vice versa.

Translation of Phrases into Expressions

Translating phrases into mathematical expressions can seem like learning a new language. However, once you crack the code, it becomes quite intuitive. To translate, first identify the key operation implied by the phrase:

For example, terms like:

• "sum" often imply addition
• "product" indicates multiplication
• "difference" implies subtraction

The exercise required us to translate "the difference of \( x \) and 2" into a mathematical expression. Here’s how:

• "Difference" directly points to the subtraction sign \(-\)
• "of \( x \) and 2" tells us which numbers or variables to subtract

Thus, the translation guides us naturally to form the expression \( x - 2 \). This skill is not only vital in prealgebra but also in more advanced math topics, ensuring students can fluidly switch between words and symbols.