Problem 6 - Intermediate Algebra

Question

Fill in the blanks. We can use the _____ property to remove parentheses in the expression \(2(x+8)\).

Step-by-Step Solution

Answer

Distributive Property

Step 1: Identify the Given Expression

The expression given is \(2(x+8)\). The task asks us to fill in the blank with a property name that allows us to remove the parentheses.

Step 2: Determine Which Property is Used

The property that allows us to distribute a factor across terms inside parentheses is called the *Distributive Property*. This property states that \(a(b+c) = ab + ac\).

Step 3: Apply the Property

Using the Distributive Property, we multiply the term outside the parentheses (2 in this case) by each term inside the parentheses: \(2 \cdot (x+8) = 2 \cdot x + 2 \cdot 8 = 2x + 16\).

Step 4: Answer the Blank

The missing property that allows us to remove parentheses in the expression \(2(x+8)\) is the Distributive Property.

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They are used to represent mathematical relationships and usually do not have an equal sign. In an algebraic expression such as \(2(x+8)\), we combine the multiplier 2 with terms inside the parentheses, which contains the variable \(x\) and the constant 8. Understanding algebraic expressions is essential as it forms the basis for solving more complex equations. These expressions offer flexibility since they can be simplified, expanded, or factorized according to the properties of operations we apply.

Some key components of algebraic expressions include:

• **Variables**: Symbols (like \(x\) or \(y\)) that represent unknown or changeable values.
• **Constants**: Fixed numbers that are part of the expression.
• **Coefficients**: Numbers multiplied with variables.
• **Terms**: Parts of the expression separated by addition or subtraction signs.

By understanding these components, one can effectively manipulate and interpret algebraic expressions.

Parentheses Removal

Removing parentheses in algebraic expressions allows simplification and can make calculations easier. The main method for removing parentheses is using mathematical properties like the Distributive Property. This process involves applying the operation outside parentheses to each term within the parentheses.

Here’s how you can remove parentheses in an expression like \(2(x+8)\):

• Identify the factor outside the parentheses (e.g., 2).
• Multiply this factor by each term inside the parentheses. In our example: \(2 imes x\) and \(2 imes 8\).
• Write out the expression as a sum: \(2x + 16\).

This method not only helps to remove parentheses but also transforms and simplifies the expression. It’s crucial for solving equations and performing algebraic manipulations.

Properties of Operations

Understanding the properties of operations provides the toolkit for manipulating and simplifying algebraic expressions. Among these, the Distributive Property is a key concept when you are dealing with expressions that involve parentheses.

Key properties include:

• **Commutative Property**: Order of addition or multiplication doesn’t affect the result (e.g., \(a + b = b + a\) or \(ab = ba\)).
• **Associative Property**: Grouping doesn’t affect the sum or product (e.g., \((a + b) + c = a + (b + c)\)).
• **Distributive Property**: Distributes a multiplicative factor over terms within parentheses (e.g., \(a(b + c) = ab + ac\)).

These properties ensure that expressions are simplified correctly and verify that mathematical procedures maintain the same sum or product. Mastery of these principles is essential for any algebra student, promoting confidence in solving problems and manipulating expressions effectively.