Which Of The Following Statements Is True?A. \[$7 - 4 = 4 - 7\$\] B. \[$7 - 4 \ \textgreater \ 4 - 7\$\] C. \[$7 - 4 \ \textless \ 4 - 7\$\] D. \[$7 - 4 = -3\$\]

by ADMIN 171 views

When it comes to mathematical expressions, the order of operations is crucial in determining the correct outcome. In this article, we will explore the given statements and determine which one is true.

The Order of Operations

The order of operations is a set of rules that dictate the order in which mathematical operations should be performed when there are 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 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 Statements

Now that we have a good understanding of the order of operations, let's evaluate the given statements.

A. ${7 - 4 = 4 - 7\$}

To evaluate this statement, we need to follow the order of operations. First, we evaluate the expressions inside the parentheses (none in this case). Then, we perform the subtraction operations.

${7 - 4 = 3\$}

${4 - 7 = -3\$}

Since ${7 - 4 \neq 4 - 7\$}, statement A is false.

B. ${7 - 4 \ \textgreater \ 4 - 7\$}

To evaluate this statement, we need to follow the order of operations. First, we evaluate the expressions inside the parentheses (none in this case). Then, we perform the subtraction operations.

${7 - 4 = 3\$}

${4 - 7 = -3\$}

Since ${7 - 4 \ \textgreater \ 4 - 7\$} is false, statement B is false.

C. ${7 - 4 \ \textless \ 4 - 7\$}

To evaluate this statement, we need to follow the order of operations. First, we evaluate the expressions inside the parentheses (none in this case). Then, we perform the subtraction operations.

${7 - 4 = 3\$}

${4 - 7 = -3\$}

Since ${7 - 4 \ \textless \ 4 - 7\$} is false, statement C is false.

D. ${7 - 4 = -3\$}

To evaluate this statement, we need to follow the order of operations. First, we evaluate the expressions inside the parentheses (none in this case). Then, we perform the subtraction operations.

${7 - 4 = 3\$}

Since ${7 - 4 \neq -3\$}, statement D is false.

Conclusion

In conclusion, none of the given statements are true. The correct evaluation of the expressions is:

${7 - 4 = 3\$}

${4 - 7 = -3\$}

Therefore, the correct answer is not among the given options.

Additional Tips and Tricks

When working with mathematical expressions, it's essential to follow the order of operations to ensure accurate results. Here are some additional tips and tricks to keep in mind:

  • Use parentheses: When in doubt, use parentheses to clarify the order of operations.
  • Evaluate expressions from left to right: When there are multiple operations in an expression, evaluate them from left to right.
  • Be careful with negative numbers: Negative numbers can change the outcome of an expression. Be careful when working with negative numbers.

By following these tips and tricks, you can ensure accurate results when working with mathematical expressions.

Common Mistakes to Avoid

When working with mathematical expressions, there are several common mistakes to avoid:

  • Not following the order of operations: Failing to follow the order of operations can lead to incorrect results.
  • Not using parentheses: Failing to use parentheses can lead to ambiguity in the order of operations.
  • Not evaluating expressions from left to right: Failing to evaluate expressions from left to right can lead to incorrect results.

By avoiding these common mistakes, you can ensure accurate results when working with mathematical expressions.

Real-World Applications

The order of operations has numerous real-world applications. Here are a few examples:

  • Finance: In finance, the order of operations is crucial when calculating interest rates, investments, and other financial transactions.
  • Science: In science, the order of operations is crucial when calculating scientific formulas, such as the formula for the area of a circle.
  • Engineering: In engineering, the order of operations is crucial when calculating complex mathematical expressions, such as those used in structural analysis.

By understanding the order of operations, you can apply mathematical concepts to real-world problems and make informed decisions.

Conclusion

In this article, we will answer some frequently asked questions about the order of operations.

Q: What is the order of operations?

A: The order of operations is a set of rules that dictate the order in which mathematical operations should be performed when there are 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 multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: Why is the order of operations important?

A: The order of operations is important because it ensures that mathematical expressions are evaluated correctly. Without the order of operations, mathematical expressions can be ambiguous and lead to incorrect results.

Q: What happens if I don't follow the order of operations?

A: If you don't follow the order of operations, you may get incorrect results. For example, consider the expression: 3 + 4 × 2. If you don't follow the order of operations, you may evaluate the expression as 3 + 4 = 7, and then multiply 7 by 2, which gives 14. However, the correct result is 3 + 8 = 11.

Q: Can I use parentheses to clarify the order of operations?

A: Yes, you can use parentheses to clarify the order of operations. For example, consider the expression: 3 + 4 × 2. To clarify the order of operations, you can use parentheses: (3 + 4) × 2. This makes it clear that the addition and multiplication operations should be performed from left to right.

Q: What about exponents? Do I need to evaluate them first?

A: Yes, you need to evaluate exponents before evaluating multiplication and division operations. For example, consider the expression: 2 × 3^2. To evaluate this expression, you need to evaluate the exponent first: 3^2 = 9. Then, you can multiply 2 by 9: 2 × 9 = 18.

Q: Can I use the order of operations to simplify complex expressions?

A: Yes, you can use the order of operations to simplify complex expressions. For example, consider the expression: (2 + 3) × (4 - 2). To simplify this expression, you can follow the order of operations:

  1. Evaluate the expressions inside the parentheses: (2 + 3) = 5 and (4 - 2) = 2.
  2. Multiply the results: 5 × 2 = 10.

Q: Are there any exceptions to the order of operations?

A: Yes, there are some exceptions to the order of operations. For example, when working with fractions, you may need to evaluate the numerator and denominator separately before performing the division operation. Additionally, when working with complex numbers, you may need to evaluate the real and imaginary parts separately before performing the addition and subtraction operations.

Q: Can I use the order of operations to evaluate expressions with multiple operations?

A: Yes, you can use the order of operations to evaluate expressions with multiple operations. For example, consider the expression: 3 + 4 × 2 - 1. To evaluate this expression, you can follow the order of operations:

  1. Evaluate the multiplication operation: 4 × 2 = 8.
  2. Add 3 and 8: 3 + 8 = 11.
  3. Subtract 1 from 11: 11 - 1 = 10.

Conclusion

In conclusion, the order of operations is a crucial concept in mathematics that dictates the order in which mathematical operations should be performed. By following the order of operations, you can ensure accurate results when working with mathematical expressions. Remember to use parentheses, evaluate exponents before multiplication and division operations, and be careful with fractions and complex numbers. By avoiding common mistakes and applying mathematical concepts to real-world problems, you can make informed decisions and achieve success in your personal and professional life.