Problem 39 - Introductory and Intermediate Algebra for College Students 4th

Question

Use a form of the distributive property to rewrite each algebraic expression without parentheses. $$\frac{1}{2}(5 x-12)$$

Step-by-Step Solution

Answer

The expression without parentheses is \(\frac{5}{2}x - 6\)

Step 1: Identify the elements a, b, and c in the given expression

In the given expression, \(\frac{1}{2}(5 x-12)\), \(\frac{1}{2}\) is 'a', '5x' is 'b' and '12' is 'c'.

Step 2: Apply the distributive property

Applying the formula a(b - c) = ab - ac, from step 1, one gets \(\frac{1}{2} * 5x - \frac{1}{2} *12\). This expands out the brackets in the equation.

Step 3: Simplify the expression

On multiplying, the expression simplifies to \(\frac{5}{2}x - 6\).

Key Concepts

Algebraic Expression Simplification

When tackling algebraic expressions, the goal is to break them down into simpler forms. Simplification makes it easier to understand the overall structure of the problem and to carry out further mathematical operations.

In our exercise, we deal with simplifying \(\frac{1}{2}(5x-12)\) by eliminating the parentheses. To do this, one must adhere to the hierarchy of operations — addressing multiplication and division before addition and subtraction. At its crux, simplification allows you to reduce an expression to its most basic terms, making it more readable and manageable for future computations.

Key steps in simplifying an algebraic expression typically include:

• Combining like terms (i.e., those with the same variable to the same power)
• Applying the distributive property
• Carrying out arithmetic operations
• Reducing fractions to their lowest terms

By following these steps, the given expression was successfully converted to a simplified form: \(\frac{5}{2}x - 6\).

Applying Distributive Property

The distributive property is a critical concept when working with algebraic expressions, acting as a bridge to simplify equations. In essence, it distributes a multiplication operation over addition or subtraction within parentheses.

Mathematically, the property is represented as \( a(b + c) = ab + ac \) or \( a(b - c) = ab - ac \). For our exercise, we applied this property to distribute \(\frac{1}{2}\) over \(5x\) and \(12\), which transformed \(\frac{1}{2}(5 x-12)\) to \(\frac{1}{2} * 5x - \frac{1}{2} *12\).

The power of the distributive property lies not only in its ability to simplify terms but also in its utility across various fields of mathematics, highlighting its omnipresence in algebraic manipulation. Its mastery is therefore fundamental for students to progress in algebra.

Intermediate Algebra

Intermediate Algebra acts as a bridge between the basic concepts learned in introductory algebra and the more complex topics that await in advanced mathematics. It encompasses a broad range of topics, one of which includes the manipulation and simplification of algebraic expressions.

The exercise displayed shows a practical application of intermediate algebra through the use of the distributive property. This property and the ability to simplify expressions are foundational tools in this field.

As students delve further into intermediate algebra, they will encounter quadratic equations, rational expressions, systems of equations, and much more. Each topic will build on the understanding that comes from applying the distributive property and simplifying expressions, demonstrating how foundational concepts in algebra are consistently applicable to more advanced problems.