Problem 4 - Prealgebra

Question

Write each of the following English phrases in symbols using the variable \(x\). The sum of three times \(x\) and 4

Step-by-Step Solution

Answer

The phrase is written as \(3x + 4\).

Step 1: Understanding the phrase

The phrase given is 'the sum of three times \(x\) and 4'. To break this down, we need to identify the components involved in the phrase. 'Three times \(x\)' suggests a multiplication operation and is represented as \(3x\). The phrase 'and 4' indicates that this number is added to something.

Step 2: Translating to symbols

Once we have identified the individual parts of the phrase, we can translate the entire English statement into a mathematical expression. 'The sum of three times \(x\) and 4' can be expressed as \(3x + 4\). Here, \(3x\) represents 'three times \(x\)' and the addition of 4 follows directly from 'and 4'.

Key Concepts

Mathematical Phrases

Mathematical phrases are similar to regular language phrases, but instead of conveying a narrative or instruction, they describe operations and relationships using mathematical expressions. When you see a statement like "the sum of three times \(x\) and 4", it's essentially a coded message describing a specific mathematical operation. This type of phrase involves:

• Numbers: These could be given directly (like '4') or indirectly (like 'three times a number').
• Variables: Symbols that represent unknown values, often letters like \(x\), which can vary.
• Operations: Actions like addition, subtraction, multiplication, or division signified by words such as 'sum', 'difference', 'product', or 'quotient'.

Understanding a mathematical phrase means breaking it down into these components. Each part of the phrase corresponds to a part of a mathematical expression. Once you can identify these parts, converting the phrase into an equation becomes more straightforward.

Variable Expressions

A variable expression is a mathematical expression that includes at least one variable. In our case, the variable is \(x\). Variables are placeholders that can represent any number, making expressions flexible and widely applicable. When you write something like 'three times \(x\)', you are talking about a variable multiplied by a constant. Here's how these components work together:

• Variables: Symbols like \(x\) that can stand in for any number.
• Coefficients: Numbers that multiply the variable, seen here as the 'three' in 'three times \(x\)'.
• Operators: Symbols denoting operations, such as \(+\) for addition or \(-\) for subtraction.

Variable expressions are crucial because they allow us to create general formulas that can solve multiple instances of a problem just by substituting different values for the variables.

Addition and Multiplication in Expressions

Addition and multiplication are fundamental operations in creating expressions. When interpreting phrases like 'the sum of three times \(x\) and 4', understanding where and how to apply addition and multiplication is key.

Multiplication

Multiplication in expressions is often indicated by terms like 'times' or 'product'. Here, 'three times \(x\)' translates into the multiplication expression \(3x\). It shows that the variable \(x\) is multiplied by the coefficient 3.

Addition

Addition is straightforward and is one of the first operations one learns in arithmetic. In expression-building, addition is commonly indicated by words like 'plus', 'added to', or 'and'. In our example, 'and 4' indicates you are adding 4 to what you already have, resulting in the expression \(3x + 4\).These operations are combined in expressions to convey more complex relationships, allowing us to solve for unknowns and understand mathematical relationships better.