What Is The Simplified Expression For The Expression Below?$\[ -1(2x + 3) - 2(x - 1) \\]A. \[$-4x + 1\$\] B. \[$-4x - 2\$\] C. \[$-4x + 2\$\] D. \[$-4x - 1\$\]

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will explore the process of simplifying a given algebraic expression, using the expression −1(2x+3)−2(x−1){-1(2x + 3) - 2(x - 1)} as an example. We will break down the expression into smaller parts, apply the distributive property, and combine like terms to arrive at the simplified expression.

Understanding the Expression

The given expression is −1(2x+3)−2(x−1){-1(2x + 3) - 2(x - 1)}. This expression consists of two parts: −1(2x+3){-1(2x + 3)} and −2(x−1){-2(x - 1)}. To simplify the expression, we need to apply the distributive property to each part.

Applying 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

We can apply this property to each part of the expression:

−1(2x+3)=−1⋅2x−1⋅3=−2x−3{-1(2x + 3) = -1 \cdot 2x -1 \cdot 3 = -2x - 3}

−2(x−1)=−2⋅x+2⋅1=−2x+2{-2(x - 1) = -2 \cdot x + 2 \cdot 1 = -2x + 2}

Combining Like Terms

Now that we have applied the distributive property to each part of the expression, we can combine like terms. Like terms are terms that have the same variable raised to the same power.

In this case, we have two terms with the variable x: −2x{-2x} and −2x{-2x}. We can combine these terms by adding their coefficients:

−2x−2x=−4x{-2x - 2x = -4x}

We also have two constant terms: −3{-3} and 2{2}. We can combine these terms by adding them:

−3+2=−1{-3 + 2 = -1}

Simplifying the Expression

Now that we have combined like terms, we can simplify the expression by combining the results:

−2x−3−2x+2=−4x−1{-2x - 3 - 2x + 2 = -4x - 1}

Conclusion

In this article, we have simplified the algebraic expression −1(2x+3)−2(x−1){-1(2x + 3) - 2(x - 1)} using the distributive property and combining like terms. The simplified expression is −4x−1{-4x - 1}. This expression is the correct answer among the options provided.

Final Answer

The final answer is −4x−1{-4x - 1}.

Why is this Important?

Simplifying algebraic expressions is an essential skill in mathematics, and it has numerous applications in various fields, including physics, engineering, and economics. By mastering this skill, you can solve complex problems and make informed decisions.

Common Mistakes to Avoid

When simplifying algebraic expressions, it's essential to avoid common mistakes, such as:

  • Forgetting to apply the distributive property
  • Failing to combine like terms
  • Making errors when adding or subtracting coefficients

By being aware of these common mistakes, you can avoid them and arrive at the correct simplified expression.

Real-World Applications

Simplifying algebraic expressions has numerous real-world applications, including:

  • Solving systems of equations
  • Finding the maximum or minimum value of a function
  • Modeling population growth or decline
  • Analyzing the behavior of complex systems

By mastering the skill of simplifying algebraic expressions, you can apply it to real-world problems and make informed decisions.

Conclusion

Introduction

In our previous article, we explored the process of simplifying algebraic expressions using the expression −1(2x+3)−2(x−1){-1(2x + 3) - 2(x - 1)} as an example. In this article, we will answer some frequently asked questions about simplifying algebraic expressions.

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that states that for any real numbers a, b, and c, the following equation holds:

a(b + c) = ab + ac

This property allows us to expand expressions by multiplying each term inside the parentheses by the factor outside the parentheses.

Q: How do I apply the distributive property?

A: To apply the distributive property, simply multiply each term inside the parentheses by the factor outside the parentheses. For example, if we have the expression −1(2x+3){-1(2x + 3)}, we can apply the distributive property by multiplying each term inside the parentheses by −1{-1}:

−1(2x+3)=−1⋅2x−1⋅3=−2x−3{-1(2x + 3) = -1 \cdot 2x -1 \cdot 3 = -2x - 3}

Q: What are like terms?

A: Like terms are terms that have the same variable raised to the same power. For example, 2x{2x} and 3x{3x} are like terms because they both have the variable x raised to the power of 1. Similarly, 2y2{2y^2} and 3y2{3y^2} are like terms because they both have the variable y raised to the power of 2.

Q: How do I combine like terms?

A: To combine like terms, simply add or subtract their coefficients. For example, if we have the expression 2x+3x{2x + 3x}, we can combine the like terms by adding their coefficients:

2x+3x=5x{2x + 3x = 5x}

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

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

  • Forgetting to apply the distributive property
  • Failing to combine like terms
  • Making errors when adding or subtracting coefficients
  • Not simplifying expressions fully

Q: How do I know when an expression is fully simplified?

A: An expression is fully simplified when there are no like terms left to combine and no further simplification can be made. For example, the expression 2x+3x+4{2x + 3x + 4} is fully simplified because there are no like terms left to combine and no further simplification can be made.

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

A: Simplifying algebraic expressions has numerous real-world applications, including:

  • Solving systems of equations
  • Finding the maximum or minimum value of a function
  • Modeling population growth or decline
  • Analyzing the behavior of complex systems

Conclusion

In conclusion, simplifying algebraic expressions is a crucial skill in mathematics, and it has numerous applications in various fields. By applying the distributive property and combining like terms, you can arrive at the simplified expression. Remember to avoid common mistakes and be aware of the real-world applications of this skill. With practice and patience, you can master the art of simplifying algebraic expressions.

Additional Resources

For more information on simplifying algebraic expressions, check out the following resources:

  • Khan Academy: Algebra
  • Mathway: Algebra
  • Wolfram Alpha: Algebra

Practice Problems

Try simplifying the following expressions:

  1. −2(3x+2)−4(x−1){-2(3x + 2) - 4(x - 1)}
  2. 3(2x−1)+2(x+2){3(2x - 1) + 2(x + 2)}
  3. −5(2x+1)+3(x−2){-5(2x + 1) + 3(x - 2)}

Answer Key

  1. −6x−4−4x+4=−10x{-6x - 4 - 4x + 4 = -10x}
  2. 6x−3+2x+4=8x+1{6x - 3 + 2x + 4 = 8x + 1}
  3. −10x−5+3x−6=−7x−11{-10x - 5 + 3x - 6 = -7x - 11}