Problem 54 Use the distributive property to... [FREE SOLUTION]

Chapter 1: Problem 54

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

Short Answer

Expert verified

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

Step by step solution

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:\[ \frac{1}{4} \times 4x + \frac{1}{4} \times (-2) \]

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 \( x - \frac{1}{2} \).

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.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

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

Short Answer

Step by step solution

Apply the Distributive Property

Distribute the Fraction

Simplify the Expression

Key Concepts

Simplifying Expressions

Algebraic Expressions

Fraction Multiplication

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Decision Maths

Probability and Statistics

Theoretical and Mathematical Physics

Statistics

Logic and Functions

Study anywhere. Anytime. Across all devices.