Which Procedure Justifies Whether − 3 X ( 5 − 4 ) + 3 ( X − 6 -3x(5-4)+3(x-6 − 3 X ( 5 − 4 ) + 3 ( X − 6 ] Is Equivalent To − 12 X − 6 -12x-6 − 12 X − 6 ?A. The Expressions Are Not Equivalent Because − 3 ( 2 ) ( 5 − 4 ) + 3 ( 2 − 6 ) = − 18 -3(2)(5-4)+3(2-6)=-18 − 3 ( 2 ) ( 5 − 4 ) + 3 ( 2 − 6 ) = − 18 And − 12 ( 2 ) − 6 = − 30 -12(2)-6=-30 − 12 ( 2 ) − 6 = − 30 .B. The Expressions Are Not Equivalent Because

In algebra, it is essential to understand the rules of equivalence and how to simplify expressions. This article will guide you through the process of determining whether two given expressions are equivalent.

Understanding the Problem

The problem presents two algebraic expressions:

1. $-3x(5-4)+3(x-6)$
2. $-12x-6$

We need to determine whether these two expressions are equivalent.

Step 1: Simplify the First Expression

To simplify the first expression, we need to follow the order of operations (PEMDAS):

1. Evaluate the expression inside the parentheses: $5-4=1$
2. Multiply $-3x$ by the result: $-3x(1)=-3x$
3. Simplify the second term: $3(x-6)=3x-18$
4. Combine the two terms: $-3x+3x-18=-18$

So, the simplified first expression is $-18$.

Step 2: Simplify the Second Expression

The second expression is already simplified: $-12x-6$.

Step 3: Compare the Two Expressions

Now that we have simplified both expressions, we can compare them:

$-18 \neq -12x-6$

The two expressions are not equivalent.

Analyzing the Incorrect Options

Let's examine the incorrect options:

A. The expressions are not equivalent because $-3(2)(5-4)+3(2-6)=-18$ and $-12(2)-6=-30$.

This option is incorrect because it uses a different value for $x$ (2) and does not represent the original expressions.

B. The expressions are not equivalent because [insert incorrect reasoning here].

This option is incomplete and does not provide a valid reason for why the expressions are not equivalent.

Conclusion

In conclusion, the two given expressions are not equivalent. The first expression simplifies to $-18$, while the second expression is $-12x-6$. The expressions are not equivalent because they have different values.

Key Takeaways

• When simplifying algebraic expressions, follow the order of operations (PEMDAS).
• Compare the simplified expressions to determine equivalence.
• Be cautious when using different values for variables, as this can lead to incorrect conclusions.

Final Answer

In this article, we will address some common questions related to the equivalence of algebraic expressions.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that dictate the order in which mathematical operations should be performed. The acronym PEMDAS stands for:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next (e.g., 2^3).
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify algebraic expressions?

A: To simplify algebraic expressions, follow these steps:

1. Evaluate any expressions inside parentheses.
2. Simplify any exponential expressions.
3. Perform any multiplication and division operations from left to right.
4. Finally, perform any addition and subtraction operations from left to right.

Q: What is the difference between equivalent and equivalent expressions?

A: Equivalent expressions are expressions that have the same value, but may be written differently. For example, the expressions $2x$ and $x+2$ are equivalent because they both represent the same value.

Q: How do I determine if two expressions are equivalent?

A: To determine if two expressions are equivalent, follow these steps:

1. Simplify both expressions using the order of operations (PEMDAS).
2. Compare the simplified expressions to determine if they are the same.

Q: What are some common mistakes to avoid when simplifying algebraic expressions?

A: Some common mistakes to avoid when simplifying algebraic expressions include:

1. Failing to evaluate expressions inside parentheses.
2. Not following the order of operations (PEMDAS).
3. Not simplifying exponential expressions.
4. Not performing multiplication and division operations from left to right.
5. Not performing addition and subtraction operations from left to right.

Q: Can you provide an example of a correct simplification of an algebraic expression?

A: Here is an example of a correct simplification of an algebraic expression:

$2x(3+2)+5$

To simplify this expression, follow the order of operations (PEMDAS):

1. Evaluate the expression inside the parentheses: $3+2=5$
2. Multiply $2x$ by the result: $2x(5)=10x$
3. Add $5$ to the result: $10x+5$

The simplified expression is $10x+5$.

Q: Can you provide an example of an incorrect simplification of an algebraic expression?

A: Here is an example of an incorrect simplification of an algebraic expression:

$2x(3+2)+5$

To simplify this expression, follow the order of operations (PEMDAS):

1. Evaluate the expression inside the parentheses: $3+2=5$
2. Multiply $2x$ by the result: $2x(5)=10x$
3. Add $5$ to the result: $10x+5$

However, if we incorrectly simplify the expression, we might get:

$2x(3+2)+5=2x(5)+5=10x+5$

This is incorrect because we failed to follow the order of operations (PEMDAS).

Conclusion

In conclusion, simplifying algebraic expressions requires careful attention to the order of operations (PEMDAS). By following these steps and avoiding common mistakes, you can ensure that your simplifications are correct.