Problem 81 - Introductory Algebra for College Students

Question

In Exercises \(81-88,\) simplify each algebraic expression by removing parentheses and brackets. $$3[5(x-2)+1]$$

Step-by-Step Solution

Answer

The simplified algebraic expression is \(15x - 27\).

Step 1: Apply Distributive Property Inside Brackets

The first operation is to apply the Distributive Property to the expression inside the brackets, which is \(5(x-2)\). This simplifies to \(5x - 10\). Combining this with \(+1\) inside the brackets, the expression becomes \(3[5x - 10 + 1]\).

Step 2: Simplify Inside Brackets

Next, simplify the expression inside the brackets. This simplifies to \(3[5x - 9]\).

Step 3: Apply Distributive Property Again

Now apply the Distributive Property again, this time to the expression inside the brackets. Multiply everything inside the brackets by the factor outside, \(3\). This simplifies to \(15x - 27\).

Key Concepts

Distributive Property

The Distributive Property is a cornerstone of algebra that allows us to multiply a single term by each term within a parenthesis. It's given by the formula: \( a(b + c) = ab + ac \). For instance, if you see an expression like \( 3(2x + 4) \), you can 'distribute' the \(3\) across the \(2x + 4\) to get \(3 \times 2x + 3 \times 4\), which simplifies to \(6x + 12\). This property is imperative when you're trying to simplify complex expressions and is a first step in removing parentheses in algebra.

Algebraic Expression Simplification

Simplifying algebraic expressions is all about making them as basic as possible. Just like simplifying fractions, you want to transform algebraic expressions into their simplest form. This involves removing parentheses, combining like terms, and performing arithmetic operations. The goal is to end up with an expression that is easier to understand and work with. It's much like cleaning and organizing a cluttered room so you can easily navigate through it. In this process, use the Distributive Property to eliminate parentheses and then combine like terms to reduce the expression to its simplest form.

Removing Parentheses in Algebra

When you remove parentheses in algebra, you’re essentially clearing the clutter to see what you really have. The Distributive Property is your primary tool in this task. Consider a packed expression like \(3[5(x - 2) + 1]\). To remove the parentheses around \(x - 2\), multiply both terms by \(5\), resulting in \(5x - 10\) while remembering to attach the \(+1\) right after. Doing so sets the stage for further simplification. After distributing the \(3\) across \(5x - 9\), the parentheses disappear, leaving a clearer, simplified expression.

Combining Like Terms

After using the Distributive Property, you may notice that some terms are similar; these are called 'like terms.' Like terms have the exact same variable raised to the same power. For example, \(2x\) and \(3x\) are like terms, but \(2x\) and \(2x^2\) are not. To combine them, simply sum up their coefficients. So, \(2x + 3x\) becomes \(5x\). This is an essential step in simplification because it reduces the number of terms in an expression, making it easier to work with. Always make sure to combine like terms after removing parentheses to achieve the simplest form.