In Exercises 121–128, write each English phrase as an algebraic expression. Then simplify the expression. Let x represent the number. The difference between the product of six and a number and negative two times the number

Verified step by step guidance

1. Step 1: Identify the key components of the English phrase. The phrase mentions 'the product of six and a number,' 'negative two times the number,' and 'the difference between' these two quantities. Let x represent the number.
2. Step 2: Translate 'the product of six and a number' into an algebraic expression. This is written as $6x$.
3. Step 3: Translate 'negative two times the number' into an algebraic expression. This is written as $-2x$.
4. Step 4: Combine these expressions to represent 'the difference between the product of six and a number and negative two times the number.' This is written as $6x\; -\; (-2x)$.
5. Step 5: Simplify the expression by resolving the subtraction of a negative term. This becomes $6x\; +\; 2x$. Combine like terms to simplify further.

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. In this context, it represents a relationship or operation involving a variable, such as 'x'. Understanding how to translate English phrases into algebraic expressions is crucial for solving algebra problems.

Product and Difference

The product refers to the result of multiplying two or more numbers or expressions, while the difference is the result of subtracting one number from another. In the given question, recognizing how to express 'the product of six and a number' and 'the difference' between two expressions is essential for forming the correct algebraic expression.

Simplification of Expressions

Simplification involves reducing an algebraic expression to its simplest form by combining like terms and performing operations. This process is important for making expressions easier to work with and understand. In this exercise, after forming the algebraic expression, simplifying it will help clarify the relationship between the variables involved.