Which Could Be Used To Evaluate The Expression -6\left(4 \frac{2}{3}\right ]?A. (-6)(4)+(-6)\left(\frac{2}{3}\right ]B. (-6)(4) \times(-6)\left(\frac{2}{3}\right ]C. (-6+4)+\left(-6+\frac{2}{3}\right ]D.

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Understanding the Problem

When evaluating expressions that involve fractions and negative numbers, it's essential to follow the correct order of operations. In this case, we're given the expression −6(423)-6\left(4 \frac{2}{3}\right) and asked to choose the correct method for evaluating it.

The Importance of Order of Operations

The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The acronym PEMDAS is commonly used to remember the order of operations:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  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.

Evaluating the Expression

Let's break down the expression −6(423)-6\left(4 \frac{2}{3}\right) and evaluate it step by step.

Step 1: Convert the Mixed Number to an Improper Fraction

First, we need to convert the mixed number 4234 \frac{2}{3} to an improper fraction. To do this, we multiply the whole number part (4) by the denominator (3) and then add the numerator (2).

423=(4×3)+23=12+23=1434 \frac{2}{3} = \frac{(4 \times 3) + 2}{3} = \frac{12 + 2}{3} = \frac{14}{3}

Step 2: Multiply the Negative Number by the Improper Fraction

Now that we have the mixed number converted to an improper fraction, we can multiply the negative number (-6) by the improper fraction.

−6(143)=−6×143=−6×143=−843-6\left(\frac{14}{3}\right) = -6 \times \frac{14}{3} = \frac{-6 \times 14}{3} = \frac{-84}{3}

Step 3: Simplify the Result

Finally, we can simplify the result by dividing the numerator (-84) by the denominator (3).

−843=−28\frac{-84}{3} = -28

Choosing the Correct Method

Now that we've evaluated the expression, let's look at the options and choose the correct method.

Option A: (−6)(4)+(−6)(23)(-6)(4)+(-6)\left(\frac{2}{3}\right)

This option is incorrect because it doesn't follow the order of operations. We need to multiply the negative number by the improper fraction first.

Option B: (−6)(4)×(−6)(23)(-6)(4) \times(-6)\left(\frac{2}{3}\right)

This option is also incorrect because it uses the wrong operation (multiplication instead of addition) and doesn't follow the order of operations.

Option C: (−6+4)+(−6+23)(-6+4)+\left(-6+\frac{2}{3}\right)

This option is incorrect because it doesn't follow the order of operations. We need to multiply the negative number by the improper fraction first.

Option D: (−6)(423)(-6)\left(4 \frac{2}{3}\right)

This option is correct because it follows the order of operations and multiplies the negative number by the improper fraction.

The final answer is: −28\boxed{-28}

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Understanding the Problem

When evaluating expressions that involve fractions and negative numbers, it's essential to follow the correct order of operations. In this case, we're given the expression −6(423)-6\left(4 \frac{2}{3}\right) and asked to choose the correct method for evaluating it.

Q&A Session

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The acronym PEMDAS is commonly used to remember the order of operations:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  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 evaluate an expression with a negative number and a fraction?

A: To evaluate an expression with a negative number and a fraction, you need to follow the order of operations. First, convert the mixed number to an improper fraction. Then, multiply the negative number by the improper fraction. Finally, simplify the result.

Q: What is the difference between a mixed number and an improper fraction?

A: A mixed number is a combination of a whole number and a fraction, such as 4234 \frac{2}{3}. An improper fraction is a fraction where the numerator is greater than or equal to the denominator, such as 143\frac{14}{3}.

Q: How do I convert a mixed number to an improper fraction?

A: To convert a mixed number to an improper fraction, you need to multiply the whole number part by the denominator and then add the numerator. For example, to convert 4234 \frac{2}{3} to an improper fraction, you would multiply 4 by 3 and then add 2.

Q: What is the correct method for evaluating the expression −6(423)-6\left(4 \frac{2}{3}\right)?

A: The correct method for evaluating the expression −6(423)-6\left(4 \frac{2}{3}\right) is to convert the mixed number to an improper fraction, multiply the negative number by the improper fraction, and then simplify the result.

Q: What is the final answer to the expression −6(423)-6\left(4 \frac{2}{3}\right)?

A: The final answer to the expression −6(423)-6\left(4 \frac{2}{3}\right) is -28.

Conclusion

Evaluating expressions with fractions and negative numbers requires following the correct order of operations. By converting mixed numbers to improper fractions, multiplying negative numbers by fractions, and simplifying the result, you can accurately evaluate expressions and arrive at the correct answer.

The final answer is: −28\boxed{-28}