Solve For { X $} . . . { \frac{x}{x+6} - 4 = \frac{-1}{x+6} \}

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Introduction

Solving for x in a complex equation can be a challenging task, especially when dealing with fractions and variables in the denominator. In this article, we will explore how to solve for x in the equation xx+6−4=−1x+6\frac{x}{x+6} - 4 = \frac{-1}{x+6} using algebraic techniques.

Understanding the Equation

The given equation is a rational equation, which means it contains fractions with variables in the numerator and denominator. To solve for x, we need to isolate the variable x on one side of the equation. The equation can be rewritten as:

xx+6−4(x+6)x+6=−1x+6\frac{x}{x+6} - \frac{4(x+6)}{x+6} = \frac{-1}{x+6}

Combining Like Terms

To simplify the equation, we can combine like terms on the left-hand side. We can start by multiplying the numerator and denominator of the second fraction by (x+6):

xx+6−4(x+6)x+6=−1x+6\frac{x}{x+6} - \frac{4(x+6)}{x+6} = \frac{-1}{x+6}

x−4(x+6)x+6=−1x+6\frac{x - 4(x+6)}{x+6} = \frac{-1}{x+6}

Simplifying the Equation

Now, we can simplify the numerator of the left-hand side by distributing the negative sign:

x−4x−24x+6=−1x+6\frac{x - 4x - 24}{x+6} = \frac{-1}{x+6}

−3x−24x+6=−1x+6\frac{-3x - 24}{x+6} = \frac{-1}{x+6}

Canceling Out Common Factors

Since the denominators are the same, we can cancel out the common factor (x+6) on both sides of the equation:

−3x−24=−1-3x - 24 = -1

Solving for x

Now, we can solve for x by isolating the variable on one side of the equation. We can start by adding 24 to both sides:

−3x=23-3x = 23

Dividing Both Sides

Next, we can divide both sides by -3 to solve for x:

x=23−3x = \frac{23}{-3}

x=−233x = -\frac{23}{3}

Conclusion

In this article, we have solved for x in the complex equation xx+6−4=−1x+6\frac{x}{x+6} - 4 = \frac{-1}{x+6} using algebraic techniques. We started by combining like terms, simplifying the equation, canceling out common factors, and finally solving for x. The solution to the equation is x = -\frac{23}{3}.

Tips and Tricks

  • When dealing with rational equations, it's essential to simplify the equation by combining like terms and canceling out common factors.
  • To solve for x, isolate the variable on one side of the equation by adding, subtracting, multiplying, or dividing both sides by the same value.
  • When dividing both sides by a negative value, make sure to change the sign of the solution.

Real-World Applications

Solving for x in complex equations has numerous real-world applications in fields such as physics, engineering, and economics. For example, in physics, solving for x can help us determine the position of an object in a given time frame. In engineering, solving for x can help us design and optimize systems such as bridges and buildings. In economics, solving for x can help us model and analyze complex economic systems.

Common Mistakes

  • When dealing with rational equations, it's easy to make mistakes by not simplifying the equation or canceling out common factors.
  • To avoid these mistakes, make sure to carefully simplify the equation and cancel out common factors before solving for x.
  • When dividing both sides by a negative value, make sure to change the sign of the solution.

Final Thoughts

Solving for x in complex equations requires patience, persistence, and practice. By following the steps outlined in this article, you can develop the skills and confidence to tackle even the most challenging equations. Remember to simplify the equation, cancel out common factors, and solve for x by isolating the variable on one side of the equation. With practice and dedication, you can become a master of solving for x in complex equations.

Introduction

Solving for x in a complex equation can be a challenging task, but with the right techniques and strategies, it can be achieved. In this article, we will answer some of the most frequently asked questions about solving for x in a complex equation.

Q: What is a complex equation?

A: A complex equation is an equation that contains fractions, variables, and constants. It can be a rational equation, a polynomial equation, or a trigonometric equation.

Q: How do I simplify a complex equation?

A: To simplify a complex equation, you need to combine like terms, cancel out common factors, and isolate the variable on one side of the equation. You can start by multiplying the numerator and denominator of each fraction by the same value to eliminate the fractions.

Q: What is the difference between a rational equation and a polynomial equation?

A: A rational equation is an equation that contains fractions, while a polynomial equation is an equation that contains variables and constants raised to powers. For example, the equation xx+6−4=−1x+6\frac{x}{x+6} - 4 = \frac{-1}{x+6} is a rational equation, while the equation x2+4x+4=0x^2 + 4x + 4 = 0 is a polynomial equation.

Q: How do I solve for x in a rational equation?

A: To solve for x in a rational equation, you need to isolate the variable on one side of the equation by adding, subtracting, multiplying, or dividing both sides by the same value. You can start by combining like terms, canceling out common factors, and simplifying the equation.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation that contains a variable raised to the power of 1, while a quadratic equation is an equation that contains a variable raised to the power of 2. For example, the equation 2x+3=52x + 3 = 5 is a linear equation, while the equation x2+4x+4=0x^2 + 4x + 4 = 0 is a quadratic equation.

Q: How do I solve for x in a quadratic equation?

A: To solve for x in a quadratic equation, you need to isolate the variable on one side of the equation by adding, subtracting, multiplying, or dividing both sides by the same value. You can start by factoring the equation, using the quadratic formula, or completing the square.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that is used to solve quadratic equations. It is given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

where a, b, and c are the coefficients of the quadratic equation.

Q: How do I use the quadratic formula to solve for x?

A: To use the quadratic formula to solve for x, you need to plug in the values of a, b, and c into the formula and simplify. You can start by plugging in the values of a, b, and c into the formula and then simplifying the expression.

Q: What are some common mistakes to avoid when solving for x in a complex equation?

A: Some common mistakes to avoid when solving for x in a complex equation include:

  • Not simplifying the equation
  • Not canceling out common factors
  • Not isolating the variable on one side of the equation
  • Not using the correct formula or technique
  • Not checking the solution for extraneous solutions

Q: How can I practice solving for x in complex equations?

A: You can practice solving for x in complex equations by working on problems and exercises in a textbook or online resource. You can also try solving for x in real-world problems and applications.

Q: What are some real-world applications of solving for x in complex equations?

A: Some real-world applications of solving for x in complex equations include:

  • Physics: Solving for x can help us determine the position of an object in a given time frame.
  • Engineering: Solving for x can help us design and optimize systems such as bridges and buildings.
  • Economics: Solving for x can help us model and analyze complex economic systems.

Q: How can I improve my skills in solving for x in complex equations?

A: You can improve your skills in solving for x in complex equations by:

  • Practicing regularly
  • Working on problems and exercises in a textbook or online resource
  • Trying to solve for x in real-world problems and applications
  • Seeking help from a teacher or tutor
  • Reviewing and practicing the formulas and techniques used to solve for x in complex equations.