**Title: Understanding Equivalent Expressions****Objective: For non-zero denominators, determine which expression is equivalent to $ \frac{1}{a} - \frac{2a}{a+1} $.**The problem provides a mathematical expression and asks to identify which option among the provided choices is equivalent to the given expression. The expression in question is:\[ \frac{1}{a} - \frac{2a}{a+1} \]### Options1. \[ \frac{-2a^3 + 1}{a(a + 1)} \]2. \[ 1 - \frac{2a}{1} \]3. \[ \frac{1 - 2a}{a(a + 1)} \]4. \[ \frac{-2a^2 + a + 1}{a(a + 1)} \]The task is to verify which of the four options correctly simplifies to the given expression considering non-zero denominators.### Detailed Analysis of Each Option- **Option 1:** $\frac{-2a^3 + 1}{a(a + 1)}$ Analyze the numerator and denominator components to see if it matches the given algebraic expression when simplified. - **Option 2:** $1 - \frac{2a}{1}$ This option simplifies straightforwardly to $1 - 2a$. Compare this with the given expression to verify its equivalence. - **Option 3:** $\frac{1 - 2a}{a(a + 1)}$ Here, assess both the numerator $(1 - 2a)$ and the denominator $(a(a + 1))$, and compare it against the target expression. - **Option 4:** $\frac{-2a^2 + a + 1}{a(a + 1)}$ Break down the numerator $(-2a^2 + a + 1)$ and match it with both components of the target expression when combined over the common denominator.By determining the steps each option undergoes to see if they simplify down to $\frac{1}{a} - \frac{2a}{a+1}$, we'll identify the equivalent expression.### ConclusionThis exercise tests your ability to algebraically manipulate expressions, identify common denominators, and correctly simplify to determine equivalency. Through this

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Description: **Title: Understanding Equivalent Expressions****Objective: For non-zero denominators, determine which expression is equivalent to $ \frac{1}{a} - \frac{2a}{a+1} $.**The problem provides a mathematical expression and asks to identify which option

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Answered: For non-zero denominators, which of the…

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For non-zero denominators, which of the following is equivalent to 1 2a a a + 1 -2a3 +1 a(a + 1) 1- 2a 1- 2a ala + 1). -2a2+ a +1 a(a + 1)

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

**Title: Understanding Equivalent Expressions**

**Objective: For non-zero denominators, determine which expression is equivalent to $ \frac{1}{a} - \frac{2a}{a+1} $.**

The problem provides a mathematical expression and asks to identify which option among the provided choices is equivalent to the given expression. The expression in question is:

\[ \frac{1}{a} - \frac{2a}{a+1} \]

### Options

1. \[ \frac{-2a^3 + 1}{a(a + 1)} \]
2. \[ 1 - \frac{2a}{1} \]
3. \[ \frac{1 - 2a}{a(a + 1)} \]
4. \[ \frac{-2a^2 + a + 1}{a(a + 1)} \]

The task is to verify which of the four options correctly simplifies to the given expression considering non-zero denominators.

### Detailed Analysis of Each Option

- **Option 1:** $\frac{-2a^3 + 1}{a(a + 1)}$

Analyze the numerator and denominator components to see if it matches the given algebraic expression when simplified.

- **Option 2:** $1 - \frac{2a}{1}$

This option simplifies straightforwardly to $1 - 2a$. Compare this with the given expression to verify its equivalence.

- **Option 3:** $\frac{1 - 2a}{a(a + 1)}$

Here, assess both the numerator $(1 - 2a)$ and the denominator $(a(a + 1))$, and compare it against the target expression.

- **Option 4:** $\frac{-2a^2 + a + 1}{a(a + 1)}$

Break down the numerator $(-2a^2 + a + 1)$ and match it with both components of the target expression when combined over the common denominator.

By determining the steps each option undergoes to see if they simplify down to $\frac{1}{a} - \frac{2a}{a+1}$, we'll identify the equivalent expression.

### Conclusion

This exercise tests your ability to algebraically manipulate expressions, identify common denominators, and correctly simplify to determine equivalency. Through this

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