Problem 15 - Algebra 1

Question

Write the verbal phrase as an algebraic expression. Use \(x\) for the variable in your expression. Three more than half of a number

Step-by-Step Solution

Answer

The algebraic expression is \(x/2 + 3\).

Step 1: Identify the number

Identify the number referred to in the verbal phrase. In this case, the phrase 'a number' is used, which is represented by variable \(x\).

Step 2: Understand the actions

Understand what is being done to that number. It's stated that half of the number is being considered, which translates to \(x/2\) in algebraic terms.

Step 3: Combine the actions

Combine the actions indicated in the phrase. In this case, the phrase 'three more than' means that 3 is being added. So, to the term \(x/2\), 3 will be added, creating the final algebraic expression \(x/2 + 3\).

Key Concepts

Verbal Phrase to Algebraic Expression

When translating a verbal phrase into an algebraic expression, it's like deciphering a code to find the mathematical equivalent. You'll mostly start with keywords such as 'more than,' 'less than,' 'the product of,' and 'divided by.' Each of these phrases signifies specific arithmetic operations.
In the problem statement, "three more than half of a number," each part of the phrase gives a clue:

• 'A number' suggests the use of a variable, often denoted as \(x\) in algebra.
• 'Half of a number' indicates dividing the variable by 2, which is written as \(x/2\).
• 'Three more than' means adding 3 to whatever comes before it. Hence, add 3 after finding \(x/2\).

This method allows you to convert understandable phrases into algebraic work, bridging verbal and numeric worlds. Through practice, recognizing these expressions becomes second nature.

Variables in Algebra

Variables in algebra serve as placeholders for numbers we might not know yet. They are represented by symbols like \(x\), \(y\), or other letters.
Imagine variables as empty boxes that can hold different values depending on the context of the expression they are a part of. By allowing us to express math problems without fixed numbers, they become very powerful tools in both simple and complex equations.
In the example "half of a number," \(x\) is our unknown number. By taking half of \(x\) (written as \(x/2\)), we begin to work mathematically without knowing \(x\)'s exact value. This flexibility is critical in algebra, highlighting one of the subject's most important skills: solving equations based on variable manipulation.

Arithmetic Operations in Algebra

Arithmetic operations in algebra use the same principles as basic arithmetic but apply them to variables and expressions.
Let's break down the key operations:

• **Addition:** Joining two amounts. Ex: \(x + 3\) adds 3 to a number \(x\).
• **Subtraction:** Removing value. Ex: \(x - 3\) deducts 3 from \(x\).
• **Multiplication:** Repeated addition or combining numbers. Ex: \(3x\) means three times \(x\).
• **Division:** Splitting into equal parts. Ex: \(x/2\) halves \(x\).

Understanding these operations through an algebraic lens is crucial in solving expressions and equations. They help discern the relationships between numbers and variables, formulating expressions that describe real-world situations. The exercise highlights how expressions like \(x/2 + 3\) result from combining these fundamental operations.