Which Expression Is Equivalent To $-4-9$?A. $-4 + (-9$\] B. $4 - 9$ C. $4 + 9$ D. $-4 + 9$

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Understanding the Problem

When dealing with mathematical expressions, it's essential to understand the order of operations and how to apply them correctly. In this case, we're given the expression −4−9-4-9 and asked to find an equivalent expression from the given options.

The 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:

  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.

Applying the Order of Operations

In the expression −4−9-4-9, we don't have any parentheses, exponents, multiplication, or division. Therefore, we can simply apply the subtraction operation.

Evaluating the Expression

To evaluate the expression −4−9-4-9, we need to subtract 9 from -4. This can be done by adding the opposite of 9, which is -9, to -4.

Finding the Equivalent Expression

Now, let's look at the given options and find the one that is equivalent to −4−9-4-9.

Option A: −4+(−9)-4 + (-9)

This option is equivalent to −4−9-4-9 because we are adding the opposite of 9, which is -9, to -4.

Option B: 4−94 - 9

This option is not equivalent to −4−9-4-9 because we are subtracting 9 from 4, not -4.

Option C: 4+94 + 9

This option is not equivalent to −4−9-4-9 because we are adding 9 to 4, not -4.

Option D: −4+9-4 + 9

This option is not equivalent to −4−9-4-9 because we are adding 9 to -4, not subtracting 9 from -4.

Conclusion

Based on our analysis, the correct answer is Option A: −4+(−9)-4 + (-9). This option is equivalent to −4−9-4-9 because we are adding the opposite of 9, which is -9, to -4.

Final Answer

The final answer is Option A: −4+(−9)-4 + (-9).

Understanding Equivalent Expressions

Equivalent expressions are mathematical expressions that have the same value, but may be written differently. In the previous article, we discussed how to find an equivalent expression for the given expression −4−9-4-9. In this article, we'll answer some frequently asked questions about equivalent expressions.

Q: What is an equivalent expression?

A: An equivalent expression is a mathematical expression that has the same value as another expression, but may be written differently.

Q: How do I find an equivalent expression?

A: To find an equivalent expression, you can use the order of operations (PEMDAS) and apply the same operations to the given expression. You can also use algebraic properties, such as the distributive property, to rewrite the expression.

Q: What are some common equivalent expressions?

A: Some common equivalent expressions include:

  • a+b=b+aa + b = b + a
  • a−b=−(b−a)a - b = - (b - a)
  • a⋅b=b⋅aa \cdot b = b \cdot a
  • a÷b=b÷aa \div b = b \div a

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

A: To determine if two expressions are equivalent, you can use the following steps:

  1. Simplify both expressions using the order of operations (PEMDAS).
  2. Compare the simplified expressions to see if they are the same.
  3. If the expressions are the same, then they are equivalent.

Q: Can I use equivalent expressions to solve equations?

A: Yes, you can use equivalent expressions to solve equations. By rewriting an equation using an equivalent expression, you can make it easier to solve.

Q: What are some real-world applications of equivalent expressions?

A: Equivalent expressions have many real-world applications, including:

  • Algebra: Equivalent expressions are used to solve equations and inequalities.
  • Calculus: Equivalent expressions are used to find derivatives and integrals.
  • Physics: Equivalent expressions are used to describe physical systems and solve problems.
  • Engineering: Equivalent expressions are used to design and optimize systems.

Q: Can I use equivalent expressions to simplify complex expressions?

A: Yes, you can use equivalent expressions to simplify complex expressions. By rewriting a complex expression using an equivalent expression, you can make it easier to evaluate.

Q: How do I know when to use equivalent expressions?

A: You should use equivalent expressions when:

  • You need to simplify a complex expression.
  • You need to solve an equation or inequality.
  • You need to describe a physical system or solve a problem.
  • You need to optimize a system or design a new one.

Conclusion

Equivalent expressions are an essential concept in mathematics, and they have many real-world applications. By understanding how to find and use equivalent expressions, you can simplify complex expressions, solve equations and inequalities, and describe physical systems. Remember to use equivalent expressions when you need to simplify a complex expression, solve an equation or inequality, or describe a physical system.

Final Answer

The final answer is that equivalent expressions are a powerful tool for simplifying complex expressions, solving equations and inequalities, and describing physical systems. By understanding how to find and use equivalent expressions, you can become a more confident and proficient mathematician.