Problem 62 - Beginning and Intermediate Algebra

Question

Use the distributive property to write each expression without parentheses Then simplify the result. See Example 4 . \(-6(2 x+1)-1\)

Step-by-Step Solution

Answer

The expression simplifies to \(-12x - 7\).

Step 1: Apply the Distributive Property

The distributive property states that for any numbers \(a\), \(b\), and \(c\), \(a(b + c) = ab + ac\). Here, we have \(-6(2x + 1)\). Applying the distributive property, we distribute \(-6\) across each term inside the parentheses: \(-6 \cdot 2x + (-6) \cdot 1\). This results in the expression \(-12x - 6\).

Step 2: Simplify the Expression

Now, take the expression from Step 1: \(-12x - 6\), and subtract \(1\) as indicated in the original expression: \(-12x - 6 - 1\). Combine the constant terms to simplify: \(-6 - 1 = -7\). Thus, the expression becomes \(-12x - 7\).

Key Concepts

Simplifying Expressions

Simplifying expressions in algebra involves reducing complex equations or expressions into their simplest possible forms. This process allows us to better understand and work with the expressions by making them easier to handle. In algebra, simplifying often means carrying out operations such as addition, subtraction, multiplication, and division to come up with a more condensed form.
For example, in the expression \[-6(2x + 1) - 1\],we first apply the distributive property by multiplying \(-6\) with both \(2x\) and \(1\).After distributing, we obtain \(-12x - 6\).
To simplify further, we subtract the constant number outside the parentheses, \(-1\),and combine like terms such as \(-6\) and \(-1\),resulting in\(-12x - 7\).
Simplification is a vital step in making expressions easier to interpret, solve, and analyze, especially when solving equations or inequalities.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. These expressions represent values but do not contain an equality or inequality sign, unlike equations. They are essential in algebra as they provide a way to construct and convey mathematical ideas using variables to represent unknown or changing quantities.
In the expression \(-6(2x + 1) - 1\),the term\(2x + 1\)inside the parentheses is an algebraic expression itself. It combines the variable \(x\) with constants through addition, showing how quantities relate to each other in symbolic form.
Algebraic expressions are the building blocks of equations, allowing mathematicians to create complex models and formulas that describe real-world scenarios or theoretical problems.

Combining Like Terms

Combining like terms is a technique used to simplify algebraic expressions by gathering and summing terms that have the same variable raised to the same power. When we refer to 'like terms,' we mean terms that share the same variable parts. This means the coefficients (the numbers in front of the variables) can be different, but the variable parts must be the same for the terms to be combined.
In the expression \(-12x - 6 - 1\),we focus on like terms, which here are the constant terms \(-6\)and \(-1\).Both are constants (no variables), so they can be added together: \(-6 - 1 = -7\).
By combining these like terms, we reduce the expression to \(-12x - 7\),making it simpler and more straightforward to work with in further calculations. This step is crucial for solving equations, as it streamlines the problem, making further operations and solutions much easier to manage.