Problem 54 - Beginning and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The expression simplifies to \( x - \frac{1}{2} \).

Step 1: Apply the Distributive Property

The distributive property states that for any numbers a, b, and c: \[ a(b + c) = ab + ac \]In this case, we need to distribute \( \frac{1}{4} \) across the terms inside the parentheses. Therefore, distribute \( \frac{1}{4} \) across \( (4x - 2) \), which gives us:\[ \frac{1}{4} \times 4x + \frac{1}{4} \times (-2) \]

Step 2: Distribute the Fraction

Multiply \( \frac{1}{4} \) by the terms inside the parentheses individually:1. \( \frac{1}{4} \times 4x = 1x = x \)2. \( \frac{1}{4} \times (-2) = -\frac{2}{4} = -\frac{1}{2} \)This gives us the expression \( x - \frac{1}{2} \).

Step 3: Simplify the Expression

Since the terms in \( x - \frac{1}{2} \) are already simplified, no further simplification is needed. We have successfully rewritten the expression without parentheses and simplified the result.

Key Concepts

Simplifying Expressions

Simplifying an expression means rewriting it in a form that is easier to work with or understand. By using the distributive property, we can eliminate parentheses, making calculations more straightforward. In the exercise, you started with \( \frac{1}{4}(4x - 2) \), which involves both a coefficient (\( \frac{1}{4} \)) and terms within parentheses.

• Step one was to apply the distributive property, which required distributing \( \frac{1}{4} \) to both terms inside the parentheses.
• By multiplying \( \frac{1}{4} \) with each term, you effectively removed the parentheses.

After this step, you obtained the expression \( x - \frac{1}{2} \). Sometimes, further simplification can be done in algebraic expressions. However, in this instance, \( x - \frac{1}{2} \) is in its simplest form because:

• The variable \( x \) stands alone without similar terms to combine.
• \( \frac{1}{2} \) is already a simplified fraction.

Expressing equations without unnecessary elements helps streamline problem-solving and enhances comprehension.

Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and arithmetic operations. They are like sentences in algebra; instead of words, they use numbers and symbols to convey mathematical ideas. In the given exercise, the expression \( \frac{1}{4}(4x - 2) \) consists of:

• The fraction \( \frac{1}{4} \), which is the coefficient, representing a part of the terms inside parentheses.
• The term \( 4x \), which involves a variable \( x \) multiplied by 4, indicating the variable's value is scaled by 4.
• The constant term \(-2\), which is a fixed numeric value inside the expression, independent of variables.

Understanding these components helps in grasping how each part of an algebraic expression functions. Further, operations such as distribution and simplification rely on understanding these individual elements' roles and interactions in the expression. Mastery of algebraic expressions is essential in problem-solving for both strictly mathematical tasks and real-world applications.

Fraction Multiplication

Multiplying fractions can be less intimidating when you understand the process. When you multiply a fraction by a whole number or another fraction, you multiply the numerators and denominators separately. For example, in the exercise, you encountered the need to multiply \( \frac{1}{4} \) by each term inside the parentheses:

• First, you calculated \( \frac{1}{4} \times 4x \). Multiplying the numerator, 1, with 4, and the denominator, 4, with 1 results in \( 4x/4 \). Simplifying yields \( x \), as \( 4/4 \) reduces to 1.
• Next, \( \frac{1}{4} \times (-2) \) was tackled by multiplying \( 1 \times (-2) \) for the numerator and \( 4 \times 1 \) for the denominator, resulting in \( -2/4 \). Simplifying this fraction gives \( -1/2 \).

A key takeaway here is that multiplication of fractions involves straightforward multiplication across the numerators and denominators but also requires simplification after multiplication. Comprehending this process empowers students to effectively engage with and resolve similar algebra problems.