Problem 6 - Prealgebra

Question

Write each of the following English phrases in symbols using the variable \(x\). Twice the sum of \(x\) and 5

Step-by-Step Solution

Answer

The expression is \(2(x + 5)\).

Step 1: Understand the Phrase

The phrase "Twice the sum of \(x\) and 5" means we need to multiply the sum of \(x\) and 5 by 2.

Step 2: Express the Sum in Symbols

First, write the expression for the sum of \(x\) and 5. In symbols, this is \(x + 5\).

Step 3: Multiply the Sum by Two

Now, take the sum \(x + 5\) and multiply it by 2. This gives us \(2(x + 5)\).

Key Concepts

Expressions

In prealgebra, expressions are a combination of numbers, variables, and arithmetic operations. They allow us to represent ideas mathematically. An expression is not the same as an equation. It doesn't have an equality sign like an equation does. Often, expressions are used to represent quantities or perform operations on numbers and variables.
When dealing with expressions, it's important to understand the terms involved. In our exercise, the phrase "Twice the sum of \(x\) and 5" translates into the expression \(2(x + 5)\). Here, "the sum of \(x\) and 5" becomes \(x + 5\), and then it's multiplied by 2 to capture the idea of the phrase "twice."

• Terms are the individual parts in an expression separated by addition or subtraction.
• Operations tell you what to do with the terms, such as multiplying by 2 in this example.
• Expressions can be simplified or rewritten, but they don't "solve" like equations because there's no equals sign to resolve.

Understanding how to translate phrases into expressions is a key prealgebra skill that provides a foundation for more advanced types of mathematical representations.

Variables

Variables are symbols used to represent unknown or changeable values. In prealgebra, variables are often denoted by letters like \(x\), \(y\), or \(z\). They allow us to write expressions and equations that can represent a range of numbers or unknown numbers.
In our exercise, the variable \(x\) is a crucial part of the expression \(2(x + 5)\). The letter \(x\) stands for a number that we don't specifically know yet or can change. This makes our expressions and equations flexible and applicable to different situations.

• Variables can stand for a single specific number or any set of numbers.
• Using variables helps us generalize arithmetic because calculations expressed with variables can apply to many possible values of the variables.
• Variables are placeholders that we often solve for in equations, but in expressions, they help describe relationships between numbers.

Understanding variables is a fundamental concept in algebra, as they lead to formulations of equations and functions that model real-world scenarios.

Arithmetic Operations

Arithmetic operations are the basic mathematical operations we use every day: addition, subtraction, multiplication, and division. In the realm of prealgebra, understanding these operations and how they interact is essential.
In our example "twice the sum of \(x\) and 5," we use both addition and multiplication:

• Addition is shown in the sum "\((x + 5)\)," which means adding 5 to \(x\).
• Multiplication is applied as "twice," crafted as "2 times the sum \((x + 5)\)." This results in the expression \(2(x + 5)\), signifying you multiply the entire sum by 2.

Arithmetic operations follow specific rules, often referred to as the "order of operations" or "PEMDAS" (parentheses, exponents, multiplication and division (from left to right), addition and subtraction (from left to right)) to accurately solve or simplify expressions and equations.

• Addition and multiplication can be distributed and associative, meaning you can rearrange and group numbers in flexible ways to simplify calculations.
• Operations can affect entire expressions, like grouping terms in parentheses before multiplying them.
• Maintaining order and structure ensures the arithmetic expressions remain accurate and true to their intended calculation.

Grasping arithmetic operations and their properties is key to navigating mathematics successfully at any level.