Solve For All Values Of $x$:$-1-\frac{x}{x-9}=\frac{1}{x-9}$

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Introduction

In this article, we will delve into solving a complex algebraic equation involving fractions. The given equation is $-1-\frac{x}{x-9}=\frac{1}{x-9}$, and our goal is to find all possible values of $x$ that satisfy this equation. We will employ various algebraic techniques, including multiplying both sides by a common denominator, expanding and simplifying expressions, and isolating the variable $x$.

Step 1: Multiply Both Sides by the Common Denominator

To begin solving the equation, we need to eliminate the fractions. The common denominator of the fractions on both sides is $x-9$. We can multiply both sides of the equation by $x-9$ to get rid of the fractions.

(x-9)\left(-1-\frac{x}{x-9}\right) = (x-9)\left(\frac{1}{x-9}\right)

Step 2: Expand and Simplify the Expressions

Now, we can expand and simplify the expressions on both sides of the equation.

-(x-9) - x = 1

Step 3: Combine Like Terms

Next, we can combine like terms on the left-hand side of the equation.

-x + 9 - x = 1

Step 4: Simplify the Equation

Now, we can simplify the equation by combining the like terms.

-2x + 9 = 1

Step 5: Isolate the Variable $x$

To isolate the variable $x$, we need to get rid of the constant term on the left-hand side of the equation. We can do this by subtracting 9 from both sides of the equation.

-2x = -8

Step 6: Solve for $x$

Finally, we can solve for $x$ by dividing both sides of the equation by -2.

x = 4

Conclusion

In this article, we have successfully solved the complex algebraic equation $-1-\frac{x}{x-9}=\frac{1}{x-9}$ and found the value of $x$ that satisfies the equation. The solution is $x = 4$. We have employed various algebraic techniques, including multiplying both sides by a common denominator, expanding and simplifying expressions, and isolating the variable $x$.

Final Answer

The final answer is $\boxed{4}$.

Related Topics

• Solving algebraic equations
• Multiplying and dividing fractions
• Expanding and simplifying expressions
• Isolating variables

References

• Algebraic Equations
• Fractions
• Expanding and Simplifying Expressions

Further Reading

• Solving Quadratic Equations
• Solving Linear Equations
• Solving Systems of Equations

Additional Resources

• Algebraic Equations Calculator
• Fractions Calculator
• Expanding and Simplifying Expressions Calculator
  

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Introduction

In our previous article, we solved the complex algebraic equation $-1-\frac{x}{x-9}=\frac{1}{x-9}$ and found the value of $x$ that satisfies the equation. In this article, we will answer some frequently asked questions related to solving this equation.

Q: What is the first step in solving the equation $-1-\frac{x}{x-9}=\frac{1}{x-9}$?

A: The first step in solving the equation is to multiply both sides by the common denominator, which is $x-9$. This will eliminate the fractions and make it easier to solve the equation.

Q: Why do we need to multiply both sides by the common denominator?

A: We need to multiply both sides by the common denominator to eliminate the fractions. This is because the fractions on both sides of the equation have the same denominator, which is $x-9$. By multiplying both sides by $x-9$, we can get rid of the fractions and make it easier to solve the equation.

Q: What is the next step after multiplying both sides by the common denominator?

A: After multiplying both sides by the common denominator, we need to expand and simplify the expressions on both sides of the equation. This will help us to isolate the variable $x$ and solve the equation.

Q: How do we expand and simplify the expressions on both sides of the equation?

A: We can expand and simplify the expressions on both sides of the equation by combining like terms. This will help us to get rid of any unnecessary terms and make it easier to solve the equation.

Q: What is the final step in solving the equation?

A: The final step in solving the equation is to isolate the variable $x$. This can be done by subtracting the constant term from both sides of the equation and then dividing both sides by the coefficient of $x$.

Q: What is the value of $x$ that satisfies the equation?

A: The value of $x$ that satisfies the equation is $x = 4$. This can be found by following the steps outlined above and solving for $x$.

Q: Can you provide an example of how to solve a similar equation?

A: Yes, here is an example of how to solve a similar equation:

Solve the equation $-2-\frac{x}{x+3}=\frac{1}{x+3}$.

To solve this equation, we can follow the same steps as before:

1. Multiply both sides by the common denominator, which is $x+3$.
2. Expand and simplify the expressions on both sides of the equation.
3. Isolate the variable $x$ by subtracting the constant term from both sides of the equation and then dividing both sides by the coefficient of $x$.

By following these steps, we can find the value of $x$ that satisfies the equation.

Q: What are some common mistakes to avoid when solving algebraic equations?

A: Some common mistakes to avoid when solving algebraic equations include:

• Not multiplying both sides by the common denominator
• Not expanding and simplifying the expressions on both sides of the equation
• Not isolating the variable $x$
• Not checking the solution to make sure it satisfies the original equation

By avoiding these common mistakes, we can ensure that we are solving the equation correctly and finding the correct value of $x$.

Conclusion

In this article, we have answered some frequently asked questions related to solving the algebraic equation $-1-\frac{x}{x-9}=\frac{1}{x-9}$. We have also provided an example of how to solve a similar equation and discussed some common mistakes to avoid when solving algebraic equations. By following the steps outlined in this article, we can ensure that we are solving the equation correctly and finding the correct value of $x$.