Solving $-(2x+4)+x=3x-4x-4$ Determining The Solution Set

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Simplify the Equation

To begin, let’s simplify both sides of the equation:

$-(2x+4)+x = 3x-4x-4$

First, distribute the negative sign on the left side:

$-2x - 4 + x = 3x - 4x - 4$

Combine like terms on both sides. On the left side, combine $-2x$ and $x$:

$-x - 4 = 3x - 4x - 4$

On the right side, combine $3x$ and $-4x$:

$-x - 4 = -x - 4$

Now we have a simplified equation: $-x - 4 = -x - 4$. This form gives us a clearer view of the equation's nature.

Analyzing the Simplified Equation

When we look at the simplified equation $-x - 4 = -x - 4$, we observe that both sides are exactly the same. This identity indicates that any value of $x$ will satisfy the equation. To further illustrate this, let’s add $x$ to both sides:

$-x - 4 + x = -x - 4 + x$

This simplifies to:

$-4 = -4$

This statement is always true, regardless of the value of $x$. This confirms that the equation is an identity and has infinitely many solutions. Identifying identities is a key skill in algebra, helping to quickly determine the solution set of an equation.

Understanding Types of Solutions

In linear equations, there are three possible types of solutions:

1. One Solution: The equation simplifies to $x = a$, where $a$ is a constant. For example, $x = 5$ is a unique solution.
2. No Solution: The equation simplifies to a false statement, such as $0 = 1$. This means there is no value of $x$ that can satisfy the equation.
3. Infinitely Many Solutions: The equation simplifies to a true statement, such as $-4 = -4$. This means any value of $x$ will satisfy the equation.

In our case, the equation simplifies to $-4 = -4$, which falls into the third category. Recognizing these solution types is fundamental in solving various algebraic problems and is a core concept in mathematics education.

Why Infinitely Many Solutions?

The reason this equation has infinitely many solutions is that it is essentially a statement of equality that holds true irrespective of the variable $x$. The left and right sides of the equation are identical after simplification. This concept is crucial for understanding algebraic identities and their implications in solving equations.

Consider this: if we substitute any value for $x$, say $x = 0$, the equation becomes:

$-(2(0) + 4) + 0 = 3(0) - 4(0) - 4$

$-4 = -4$

This is true. If we substitute $x = 1$, the equation becomes:

$-(2(1) + 4) + 1 = 3(1) - 4(1) - 4$

$-6 + 1 = 3 - 4 - 4$

$-5 = -5$

This is also true. No matter what value we substitute for $x$, the equation will always hold true. This is the hallmark of an equation with infinitely many solutions. Testing different values of $x$ can often help to confirm the nature of the solutions.

Common Mistakes to Avoid

When solving equations, several common mistakes can lead to incorrect conclusions. Here are a few to avoid:

1. Incorrect Distribution: Failing to distribute negative signs correctly. For example, incorrectly expanding $-(2x + 4)$ as $-2x + 4$ instead of $-2x - 4$.
2. Combining Unlike Terms: Incorrectly combining terms that are not like terms. For example, combining $2x$ and $4$ as $6x$.
3. Arithmetic Errors: Making simple arithmetic mistakes while simplifying the equation. Double-checking each step can help prevent these errors.
4. Misinterpreting Results: Misinterpreting a true statement (e.g., $-4 = -4$) as no solution, or a false statement (e.g., $0 = 1$) as infinitely many solutions. Avoiding these common pitfalls can significantly improve accuracy in solving algebraic equations.

Conclusion

In summary, the equation $-(2x+4)+x=3x-4x-4$ has infinitely many solutions. This is because, after simplification, both sides of the equation are identical, resulting in a true statement regardless of the value of $x$. Understanding how to simplify equations and recognize different types of solutions is fundamental in algebra. Mastering these concepts provides a strong foundation for more advanced mathematical topics.

Therefore, the correct answer is:

D. The equation has infinitely many solutions.

This detailed explanation should help students and anyone studying algebra to grasp the concept of infinitely many solutions and how to identify them in linear equations. Continuous practice and understanding of fundamental concepts are key to success in mathematics. By simplifying the equation step-by-step and analyzing the result, we were able to correctly determine that the equation has infinitely many solutions. This process highlights the importance of careful algebraic manipulation and a solid understanding of the properties of equality. Further practice with similar problems will reinforce these skills and improve problem-solving abilities in algebra.

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Understanding the Question

In this exercise, we are tasked with solving a linear equation: $-(2x+4)+x=3x-4x-4$. The primary goal is to identify the correct statement that describes the solution set of this equation. This requires simplifying the equation and determining whether it has a unique solution, no solution, or infinitely many solutions. Effective problem-solving begins with a clear understanding of the question and the steps needed to arrive at the answer.

Detailed Step-by-Step Solution

To correctly determine the solution set, we must meticulously simplify and solve the given equation. Let's break down the process step-by-step:

Step 1: Distribute the Negative Sign

The first step involves distributing the negative sign in front of the parentheses on the left side of the equation:

$-(2x+4)+x = 3x-4x-4$

Distributing the negative sign, we get:

$-2x - 4 + x = 3x - 4x - 4$

Accurate distribution is crucial to avoid errors in the simplification process.

Step 2: Combine Like Terms on Each Side

Next, we combine like terms on both sides of the equation. On the left side, we combine $-2x$ and $+x$:

$-2x + x - 4 = 3x - 4x - 4$

This simplifies to:

$-x - 4 = 3x - 4x - 4$

On the right side, we combine $3x$ and $-4x$:

$-x - 4 = -x - 4$

Proper combination of like terms is essential for simplifying the equation.

Step 3: Analyze the Simplified Equation

After simplifying, we have the equation:

$-x - 4 = -x - 4$

Notice that both sides of the equation are identical. This means that the equation is an identity, and it will be true for any value of $x$. This indicates that the equation has infinitely many solutions. Recognizing identities is key to determining solution sets quickly and accurately.

Step 4: Confirm Infinitely Many Solutions

To further confirm, let's add $x$ to both sides of the equation:

$-x - 4 + x = -x - 4 + x$

This simplifies to:

$-4 = -4$

The statement $-4 = -4$ is always true, regardless of the value of $x$. This definitively confirms that the equation has infinitely many solutions. Verifying the result through additional steps ensures the correctness of the solution.

Evaluating the Given Options

Now that we have solved the equation, let's evaluate the given options to select the correct answer:

A. The equation has one solution, $x=0$.

• This is incorrect because the equation has infinitely many solutions, not just one.

B. The equation has one solution, $x=4$.

• This is also incorrect for the same reason as option A.

C. The equation has no solution.

• This is incorrect because the equation has infinitely many solutions, not no solution.

D. The equation has infinitely many solutions.

• This is the correct answer, as we have demonstrated through our step-by-step solution.

Therefore, the correct answer is:

D. The equation has infinitely many solutions.

Common Mistakes and How to Avoid Them

Solving linear equations can be tricky, and several common mistakes can lead to incorrect answers. Here are some to watch out for:

1. Incorrectly Distributing the Negative Sign

A common mistake is to incorrectly distribute the negative sign. For example, students might write $-(2x + 4)$ as $-2x + 4$ instead of $-2x - 4$. Always ensure the negative sign is correctly distributed to each term inside the parentheses.

2. Combining Unlike Terms Incorrectly

Another mistake is combining terms that are not like terms. For instance, combining $-2x$ and $-4$ as $-6x$. Remember to only combine terms that have the same variable and exponent.

3. Arithmetic Errors

Simple arithmetic errors can easily occur during the simplification process. Double-check each step to minimize the chances of making mistakes.

4. Misinterpreting the Result

Students might misinterpret the simplified equation. For example, getting $-4 = -4$ and thinking there is no solution, or getting $0 = 1$ and thinking there are infinitely many solutions. Understand the meaning of the final statement to correctly identify the solution set.

Practical Tips for Solving Linear Equations

To improve your skills in solving linear equations, consider these practical tips:

1. Write Each Step Clearly: Show all the steps in your solution process. This helps in identifying errors and makes it easier to follow your work.
2. Check Your Work: After solving the equation, substitute the solution back into the original equation to verify if it holds true.
3. Practice Regularly: Consistent practice is key to mastering linear equations. Solve a variety of problems to build your skills.
4. Understand the Concepts: Don't just memorize steps; understand the underlying concepts. This will help you solve more complex problems.

Conclusion

In conclusion, the equation $-(2x+4)+x=3x-4x-4$ has infinitely many solutions. This was determined by simplifying the equation step-by-step and recognizing that the final simplified form is an identity. By avoiding common mistakes and following practical tips, you can improve your ability to solve linear equations accurately and efficiently. Consistent effort and understanding are the keys to success in mathematics. The process of solving this equation underscores the importance of careful algebraic manipulation and a strong grasp of the properties of equality. Continued practice with similar problems will help solidify these skills and enhance your problem-solving capabilities in algebra.