Simplify: ${ -4(x+3) - 3(x+4) }$

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Introduction

In mathematics, simplifying algebraic expressions is a crucial skill that helps in solving equations and inequalities. It involves combining like terms and removing any unnecessary components from the expression. In this article, we will simplify the given expression $-4(x+3) - 3(x+4)$ using the distributive property and combining like terms.

Understanding the Expression

The given expression is $-4(x+3) - 3(x+4)$. This expression consists of two terms, each of which is a product of a constant and a binomial. The first term is $-4(x+3)$, and the second term is $-3(x+4)$.

Distributive Property

To simplify the expression, we will use the distributive property, which states that for any real numbers $a$, $b$, and $c$, $a(b+c) = ab + ac$. We will apply this property to each term in the expression.

Distributing the Negative 4

We will start by distributing the negative 4 to the terms inside the parentheses of the first term.

-4(x+3) = -4x - 12

Distributing the Negative 3

Next, we will distribute the negative 3 to the terms inside the parentheses of the second term.

-3(x+4) = -3x - 12

Combining Like Terms

Now that we have distributed the constants to the terms inside the parentheses, we can combine like terms. Like terms are terms that have the same variable raised to the same power.

-4x - 12 - 3x - 12 = -7x - 24

Final Simplified Expression

The final simplified expression is $-7x - 24$. This is the result of combining like terms and removing any unnecessary components from the original expression.

Conclusion

In this article, we simplified the expression $-4(x+3) - 3(x+4)$ using the distributive property and combining like terms. We started by distributing the constants to the terms inside the parentheses and then combined like terms to get the final simplified expression. This process is essential in solving equations and inequalities and is a crucial skill in mathematics.

Real-World Applications

Simplifying algebraic expressions has many real-world applications. For example, in physics, simplifying expressions is used to solve problems involving motion, energy, and forces. In engineering, simplifying expressions is used to design and optimize systems, such as electrical circuits and mechanical systems.

Tips and Tricks

Here are some tips and tricks to help you simplify algebraic expressions:

• Use the distributive property to distribute constants to terms inside parentheses.
• Combine like terms to remove unnecessary components from the expression.
• Use the order of operations (PEMDAS) to evaluate expressions.
• Simplify expressions step by step to avoid mistakes.

Common Mistakes

Here are some common mistakes to avoid when simplifying algebraic expressions:

• Failing to distribute constants to terms inside parentheses.
• Failing to combine like terms.
• Not using the order of operations (PEMDAS) to evaluate expressions.
• Not simplifying expressions step by step.

Practice Problems

Here are some practice problems to help you practice simplifying algebraic expressions:

• Simplify the expression $2(x+5) - 3(x+2)$.
• Simplify the expression $-4(x-3) + 2(x-2)$.
• Simplify the expression $3(x+2) - 2(x+1)$.

Answer Key

Here are the answers to the practice problems:

• $2(x+5) - 3(x+2) = -x + 6$
• $-4(x-3) + 2(x-2) = -6x + 10$
• $3(x+2) - 2(x+1) = x + 4$

Conclusion

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Introduction

In our previous article, we simplified the expression $-4(x+3) - 3(x+4)$ using the distributive property and combining like terms. In this article, we will answer some frequently asked questions about simplifying algebraic expressions.

Q&A

Q: What is the distributive property?

A: The distributive property is a mathematical concept that states that for any real numbers $a$, $b$, and $c$, $a(b+c) = ab + ac$. This means that we can distribute a constant to the terms inside the parentheses.

Q: How do I distribute a constant to the terms inside the parentheses?

A: To distribute a constant to the terms inside the parentheses, we multiply the constant by each term inside the parentheses. For example, if we have the expression $2(x+3)$, we would distribute the 2 to the terms inside the parentheses as follows:

2(x+3) = 2x + 6

Q: What are like terms?

A: Like terms are terms that have the same variable raised to the same power. For example, $2x$ and $5x$ are like terms because they both have the variable $x$ raised to the power of 1.

Q: How do I combine like terms?

A: To combine like terms, we add or subtract the coefficients of the like terms. For example, if we have the expression $2x + 5x$, we would combine the like terms as follows:

2x + 5x = 7x

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

A: The order of operations (PEMDAS) is a mathematical concept that states that we should perform operations in the following order:

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: Why is it important to simplify algebraic expressions?

A: Simplifying algebraic expressions is important because it helps us to:

• Solve equations and inequalities
• Evaluate expressions
• Understand the relationships between variables
• Make predictions and models in real-world applications

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

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

• Failing to distribute constants to terms inside parentheses
• Failing to combine like terms
• Not using the order of operations (PEMDAS) to evaluate expressions
• Not simplifying expressions step by step

Conclusion

In conclusion, simplifying algebraic expressions is a crucial skill in mathematics that has many real-world applications. By understanding the distributive property, combining like terms, and using the order of operations (PEMDAS), we can simplify expressions and solve problems involving motion, energy, and forces. Remember to practice simplifying expressions regularly to become proficient in this skill.

Practice Problems

Here are some practice problems to help you practice simplifying algebraic expressions:

• Simplify the expression $3(x+2) - 2(x+1)$.
• Simplify the expression $-4(x-3) + 2(x-2)$.
• Simplify the expression $2(x+5) - 3(x+2)$.

Answer Key

Here are the answers to the practice problems:

• $3(x+2) - 2(x+1) = x + 3$
• $-4(x-3) + 2(x-2) = -6x + 10$
• $2(x+5) - 3(x+2) = -x + 6$

Real-World Applications

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

• Physics: Simplifying expressions is used to solve problems involving motion, energy, and forces.
• Engineering: Simplifying expressions is used to design and optimize systems, such as electrical circuits and mechanical systems.
• Economics: Simplifying expressions is used to model and analyze economic systems.
• Computer Science: Simplifying expressions is used to develop algorithms and data structures.

Tips and Tricks

Here are some tips and tricks to help you simplify algebraic expressions:

• Use the distributive property to distribute constants to terms inside parentheses.
• Combine like terms to remove unnecessary components from the expression.
• Use the order of operations (PEMDAS) to evaluate expressions.
• Simplify expressions step by step to avoid mistakes.

Conclusion

In conclusion, simplifying algebraic expressions is a crucial skill in mathematics that has many real-world applications. By understanding the distributive property, combining like terms, and using the order of operations (PEMDAS), we can simplify expressions and solve problems involving motion, energy, and forces. Remember to practice simplifying expressions regularly to become proficient in this skill.