Problem 76 - Introductory Algebra for College Students

Question

Write each English phrase as an algebraic expression. Then simplify the expression. Let \(x\) represent the number. nine increased by the product of 3 and 2 less than a number

Step-by-Step Solution

Answer

The algebra expression that represents the phrase 'nine increased by the product of 3 and 2 less than a number' simplifies to \(3x + 3\).

Step 1: Identify and Define the Variables

The problem provides that \(x\) represent the number.

Step 2: Decompose the Statement

Let's break the statement into different parts to manage the complexity. The statement is: 'nine increased by the product of 3 and 2 less than a number'. This can be decomposed into two: 'nine increased by something' and 'the product of 3 and 2 less than a number'.

Step 3: Formulate and Combine Expressions

'Nine increased by something' can be formulated as \(9+something\). The 'something' being referred to here is 'the product of 3 and 2 less than a number'. This can be formulated as \(3*(x-2)\). Thus the whole statement is represented as \(9 + 3*(x-2)\).

Step 4: Simplify Expression

We then simplify to amalgamate like terms together, resulting in the algebraic expression \(9+3x - 6\) which further simplifies to \(3x + 3\).

Key Concepts

Variable Representation

In algebra, variables play a crucial role. They act as placeholders for numbers we do not know yet or numbers that can change. In our exercise, we have been given that the variable \(x\) represents 'the number'. This is where variable representation comes in.

By assigning \(x\) to represent the unknown number, we set ourselves up to translate words into mathematical language.

• "2 less than a number" translates to \(x-2\).
• "Product of 3 and 2 less than a number" becomes \(3*(x-2)\).

This process of replacing words with corresponding algebraic expressions using variables is foundational in solving algebraic problems.

Expression Simplification

Expression simplification is the process of combining and reducing algebraic expressions into their simplest form. Once we have our expression, like \(9 + 3*(x-2)\), the next step is to simplify it. Simplification often involves a few consistent techniques.

• Distribute any numbers outside of parentheses throughout the terms inside the parentheses. In our expression: \(3*(x-2) = 3x - 6\).
• Combine like terms by adding or subtracting them. In the example, after distribution, we have: \(9 + 3x - 6\).
• Reorder and combine to simplify further, resulting in: \(3x + 3\).

This step helps create a clearer, more manageable expression that is easier to work with in further mathematical operations.

Product of Numbers

The concept of a product in mathematics refers to the result of multiplying two or more numbers together. In this exercise, there is a specific need to find the product of 3 and "2 less than a number".

Here's how we approach it:

• First, identify the numbers or expressions to be multiplied together. Here they are 3 and \(x-2\) (which represents 2 less than \(x\)).
• Perform the multiplication, often referred to as distribution when it involves expressions. Thus, \(3*(x-2)\) is distributed to become \(3x - 6\).

The concept of multiplication, when connected to variables and subtracted terms, often involves thoughtful application of distributive properties and proper arithmetic to yield the correct algebraic product.