Problem 32 - Algebra 1

Question

DISTRIBUTIVE PROPERTY Use the distributive property to rewrite the expression without parentheses. $$ (x+6)(-2) $$

Step-by-Step Solution

Answer

The expression \( (x+6)(-2) \), rewritten without parentheses using the distributive property, is \( -2x - 12 \)

Step 1: Identify Outer Term

Firstly, identify the term outside the bracket, which is -2 in this case.

Step 2: Apply Distributive Property Part - 1

Then apply the distributive property by multiplying -2 with the first term in the bracket, which is x, giving us -2x.

Step 3: Apply Distributive Property Part - 2

Now, multiply -2 with the second term in the bracket, which is +6, giving us -12.

Step 4: Write Down Final Expression

Combine both products to write down the final expression without parentheses. The combined expression is -2x - 12.

Key Concepts

Algebraic Expressions

Algebraic expressions are the cornerstone of algebra and represent a combination of numbers, variables, and arithmetic operations. For example, in the expression \( (x+6)(-2) \), we have variables (\( x \)), numbers (\( 6 \), \( -2 \) ), and operations (addition and multiplication). These expressions are akin to written sentences in algebra and their structure determines the meaning and the operations that can be performed on them.

Understanding how to manipulate algebraic expressions is essential for solving equations and understanding more complex mathematical concepts. Essentially, an expression provides us with a formula that describes a particular relationship between different mathematical entities. Simplifying these expressions, as seen in our given exercise, helps to clarify that relationship by combining like terms and making the expression more manageable to work with.

Simplifying Expressions

Simplifying expressions is a fundamental skill when dealing with algebraic expressions. When we simplify an expression, we're making it easier to understand and often shorter, without changing its value. The process involves combining like terms and using mathematical properties such as the distributive property to eliminate parentheses and group similar types of terms together.

For instance, the expression \( (x+6)(-2) \) can be simplified using the distributive property. By applying this property, we break down the original expression into simpler components. The distributive property allows us to multiply the -2 by each term within the parentheses, simplifying the original expression to \( -2x - 12 \) which is much easier to work with if we need to use it within a larger problem or equation.

Properties of Operations

Properties of operations are the basic rules that apply to arithmetic operations. They provide a structure and order for how we solve mathematical problems. The three most important properties are the commutative, associative, and distributive properties.

The distributive property is particularly important and is the focus of our given exercise. It states that for any numbers a, b, and c, the expression \( a(b + c) \) is equal to \( ab + ac \). In essence, this property allows us to multiply a number by a group of numbers added together by multiplying individually and then adding the results. This is exactly what we applied in the exercise to simplify \( (x+6)(-2) \), resulting in \( -2x -12 \).

Applying the distributive property correctly is crucial for simplifying expressions and solving equations efficiently. It's a powerful tool that can help students tackle algebraic expressions methodically and with confidence.