Which Expression Is Equivalent To -8\left(3x-\frac{2}{3}\right ]?A. − 2 X + 8 -2x + 8 − 2 X + 8 B. − 2 -2 − 2 C. − 18 X + 12 -18x + 12 − 18 X + 12 D. − 18 X + 4 -18x + 4 − 18 X + 4

Introduction

In algebra, the distributive property is a fundamental concept that allows us to expand and simplify expressions. It states that for any real numbers a, b, and c, the following equation holds: a(b + c) = ab + ac. In this article, we will explore how to apply the distributive property to simplify expressions, using the given problem as a case study.

The Problem

The problem asks us to find the equivalent expression for $-8\left(3x-\frac{2}{3}\right)$. To solve this problem, we will apply the distributive property to expand the expression.

Step 1: Apply the Distributive Property

The distributive property states that for any real numbers a, b, and c, the following equation holds: a(b + c) = ab + ac. In this case, we have $-8\left(3x-\frac{2}{3}\right)$, which can be rewritten as $-8(3x) + (-8)\left(-\frac{2}{3}\right)$.

Step 2: Simplify the Expression

Now that we have applied the distributive property, we can simplify the expression by multiplying the coefficients and the variables.

$-8(3x) = -24x$

$(-8)\left(-\frac{2}{3}\right) = \frac{16}{3}$

Step 3: Combine the Terms

Now that we have simplified the expression, we can combine the terms to get the final answer.

$-24x + \frac{16}{3}$

The Final Answer

The final answer is $-24x + \frac{16}{3}$. However, we need to compare this answer with the given options to determine which one is equivalent.

Comparing the Answers

Let's compare the final answer with the given options.

A. $-2x + 8$

B. $-2$

C. $-18x + 12$

D. $-18x + 4$

We can see that option D is the only one that matches the final answer.

Conclusion

In this article, we applied the distributive property to simplify the expression $-8\left(3x-\frac{2}{3}\right)$. We broke down the problem into three steps: applying the distributive property, simplifying the expression, and combining the terms. Finally, we compared the final answer with the given options to determine which one is equivalent. The correct answer is option D: $-18x + 4$.

Key Takeaways

• The distributive property is a fundamental concept in algebra that allows us to expand and simplify expressions.
• To apply the distributive property, we need to multiply the coefficients and the variables.
• We can simplify the expression by combining the terms.
• The final answer should be compared with the given options to determine which one is equivalent.

Frequently Asked Questions

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that allows us to expand and simplify expressions. It states that for any real numbers a, b, and c, the following equation holds: a(b + c) = ab + ac.

Q: How do I apply the distributive property?

A: To apply the distributive property, you need to multiply the coefficients and the variables. For example, if you have $-8\left(3x-\frac{2}{3}\right)$, you can rewrite it as $-8(3x) + (-8)\left(-\frac{2}{3}\right)$.

Q: How do I simplify the expression?

A: To simplify the expression, you need to combine the terms. For example, if you have $-24x + \frac{16}{3}$, you can combine the terms to get the final answer.

Q: How do I compare the answers?

A: To compare the answers, you need to look at the final answer and compare it with the given options. In this case, the final answer is $-24x + \frac{16}{3}$, which matches option D: $-18x + 4$.

Additional Resources

• Khan Academy: Distributive Property
• Mathway: Distributive Property
• Algebra.com: Distributive Property

Conclusion

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Introduction

The distributive property is a fundamental concept in algebra that allows us to expand and simplify expressions. In our previous article, we explored how to apply the distributive property to simplify expressions, using the given problem as a case study. In this article, we will provide a comprehensive Q&A section to help you better understand the distributive property and how to apply it in different scenarios.

Q&A Section

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that allows us to expand and simplify expressions. It states that for any real numbers a, b, and c, the following equation holds: a(b + c) = ab + ac.

Q: How do I apply the distributive property?

A: To apply the distributive property, you need to multiply the coefficients and the variables. For example, if you have $-8\left(3x-\frac{2}{3}\right)$, you can rewrite it as $-8(3x) + (-8)\left(-\frac{2}{3}\right)$.

Q: What are the steps to apply the distributive property?

A: The steps to apply the distributive property are:

1. Identify the expression that needs to be expanded.
2. Apply the distributive property by multiplying the coefficients and the variables.
3. Simplify the expression by combining the terms.

Q: How do I simplify the expression?

A: To simplify the expression, you need to combine the terms. For example, if you have $-24x + \frac{16}{3}$, you can combine the terms to get the final answer.

Q: What are some common mistakes to avoid when applying the distributive property?

A: Some common mistakes to avoid when applying the distributive property include:

• Not multiplying the coefficients and the variables correctly.
• Not simplifying the expression correctly.
• Not combining the terms correctly.

Q: How do I compare the answers?

A: To compare the answers, you need to look at the final answer and compare it with the given options. In this case, the final answer is $-24x + \frac{16}{3}$, which matches option D: $-18x + 4$.

Q: What are some real-world applications of the distributive property?

A: The distributive property has many real-world applications, including:

• Algebraic expressions in physics and engineering.
• Financial calculations, such as calculating interest rates.
• Data analysis and statistics.

Q: How do I practice the distributive property?

A: To practice the distributive property, you can try the following:

• Use online resources, such as Khan Academy or Mathway, to practice the distributive property.
• Work on algebraic expressions that involve the distributive property.
• Practice simplifying expressions using the distributive property.

Q: What are some common misconceptions about the distributive property?

A: Some common misconceptions about the distributive property include:

• Thinking that the distributive property only applies to multiplication.
• Thinking that the distributive property only applies to addition and subtraction.
• Thinking that the distributive property is only used in algebra.

Conclusion

In this article, we provided a comprehensive Q&A section to help you better understand the distributive property and how to apply it in different scenarios. We covered topics such as the definition of the distributive property, how to apply it, common mistakes to avoid, and real-world applications. We also provided some tips on how to practice the distributive property and common misconceptions to avoid.

Additional Resources

• Khan Academy: Distributive Property
• Mathway: Distributive Property
• Algebra.com: Distributive Property

Practice Problems

1. Simplify the expression $-3\left(2x + 5\right)$ using the distributive property.
2. Simplify the expression $-2\left(x - 3\right)$ using the distributive property.
3. Simplify the expression $-4\left(3x - 2\right)$ using the distributive property.

Answers

1. $-6x - 15$
2. $-2x + 6$
3. $-12x + 8$

Conclusion

In this article, we provided a comprehensive Q&A section to help you better understand the distributive property and how to apply it in different scenarios. We covered topics such as the definition of the distributive property, how to apply it, common mistakes to avoid, and real-world applications. We also provided some tips on how to practice the distributive property and common misconceptions to avoid.