Solve The Rational Equation:${ 12 - \frac{34}{x} = -4 + \frac{30}{x} }$A. { X = -0.6 $}$ B. There Is No Solution. C. { X = 4 $}$ D. { X = 8 $}$

Introduction

Rational equations are a type of algebraic equation that involves fractions and variables. They can be challenging to solve, but with the right approach, you can master the art of solving them. In this article, we will focus on solving the rational equation: $12 - \frac{34}{x} = -4 + \frac{30}{x}$. We will break down the solution step by step and provide a clear explanation of each step.

Understanding Rational Equations

A rational equation is an equation that contains one or more fractions with variables in the numerator or denominator. Rational equations can be linear or quadratic, and they can involve multiple variables. To solve a rational equation, you need to isolate the variable and eliminate the fractions.

Step 1: Eliminate the Fractions

To eliminate the fractions, we need to multiply both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the LCM of $x$ and $x$ is $x$. So, we multiply both sides of the equation by $x$.

$$x(12 - \frac{34}{x}) = x(-4 + \frac{30}{x})$$

This simplifies to:

$$12x - 34 = -4x + 30$$

Step 2: Combine Like Terms

Now, we need to combine like terms on both sides of the equation. On the left side, we have $12x$ and $-34$. On the right side, we have $-4x$ and $30$.

$$12x - 34 = -4x + 30$$

Combine like terms:

$$16x - 34 = 30$$

Step 3: Add 34 to Both Sides

To isolate the term with the variable, we need to add 34 to both sides of the equation.

$$16x - 34 + 34 = 30 + 34$$

This simplifies to:

$$16x = 64$$

Step 4: Divide Both Sides by 16

Finally, we need to divide both sides of the equation by 16 to solve for $x$.

$$\frac{16x}{16} = \frac{64}{16}$$

This simplifies to:

$$x = 4$$

Conclusion

In this article, we solved the rational equation: $12 - \frac{34}{x} = -4 + \frac{30}{x}$. We broke down the solution step by step and provided a clear explanation of each step. We eliminated the fractions, combined like terms, added 34 to both sides, and finally divided both sides by 16 to solve for $x$. The solution to the equation is $x = 4$.

Answer

The correct answer is:

• C. $x = 4$

Discussion

Rational equations can be challenging to solve, but with the right approach, you can master the art of solving them. In this article, we provided a step-by-step guide to solving rational equations. We hope that this article has been helpful in understanding how to solve rational equations.

Common Mistakes

When solving rational equations, it's easy to make mistakes. Here are some common mistakes to avoid:

• Not eliminating the fractions: Failing to eliminate the fractions can lead to incorrect solutions.
• Not combining like terms: Failing to combine like terms can lead to incorrect solutions.
• Not adding or subtracting the same value to both sides: Failing to add or subtract the same value to both sides can lead to incorrect solutions.

Tips and Tricks

Here are some tips and tricks to help you solve rational equations:

• Use the least common multiple (LCM) to eliminate fractions: The LCM is the smallest multiple that both denominators have in common.
• Combine like terms: Combine like terms on both sides of the equation to simplify the equation.
• Add or subtract the same value to both sides: Add or subtract the same value to both sides of the equation to isolate the variable.

Conclusion

Introduction

In our previous article, we provided a step-by-step guide to solving rational equations. However, we know that sometimes, it's easier to understand a concept by asking questions and getting answers. In this article, we will provide a Q&A guide to solving rational equations. We will answer common questions and provide examples to help you understand the concept.

Q: What is a rational equation?

A: A rational equation is an equation that contains one or more fractions with variables in the numerator or denominator.

Q: How do I solve a rational equation?

A: To solve a rational equation, you need to follow these steps:

1. Eliminate the fractions by multiplying both sides of the equation by the least common multiple (LCM) of the denominators.
2. Combine like terms on both sides of the equation.
3. Add or subtract the same value to both sides of the equation to isolate the variable.
4. Divide both sides of the equation by the coefficient of the variable to solve for the variable.

Q: What is the least common multiple (LCM)?

A: The least common multiple (LCM) is the smallest multiple that both denominators have in common. For example, if the denominators are 2 and 3, the LCM is 6.

Q: How do I find the LCM?

A: To find the LCM, you can list the multiples of each denominator and find the smallest multiple that they have in common. Alternatively, you can use the following formula:

LCM(a, b) = (a × b) / GCD(a, b)

where GCD(a, b) is the greatest common divisor of a and b.

Q: What is the greatest common divisor (GCD)?

A: The greatest common divisor (GCD) is the largest number that divides both numbers without leaving a remainder.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the terms with the same variable. For example, if you have the equation 2x + 3x, you can combine the terms by adding the coefficients:

2x + 3x = 5x

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

A: A rational equation is an equation that contains one or more fractions with variables in the numerator or denominator. A rational expression is an expression that contains one or more fractions with variables in the numerator or denominator.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you need to follow these steps:

1. Factor the numerator and denominator.
2. Cancel out any common factors.
3. Simplify the expression.

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

A: A rational equation is an equation that contains one or more fractions with variables in the numerator or denominator. A quadratic equation is an equation that contains a squared variable.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you need to follow these steps:

1. Factor the equation.
2. Set each factor equal to zero and solve for the variable.
3. Simplify the solutions.

Conclusion

In conclusion, solving rational equations requires a step-by-step approach. We provided a Q&A guide to help you understand the concept and provided examples to help you practice. With practice and patience, you can master the art of solving rational equations.

Common Mistakes

When solving rational equations, it's easy to make mistakes. Here are some common mistakes to avoid:

• Not eliminating the fractions: Failing to eliminate the fractions can lead to incorrect solutions.
• Not combining like terms: Failing to combine like terms can lead to incorrect solutions.
• Not adding or subtracting the same value to both sides: Failing to add or subtract the same value to both sides can lead to incorrect solutions.

Tips and Tricks

Here are some tips and tricks to help you solve rational equations:

• Use the least common multiple (LCM) to eliminate fractions: The LCM is the smallest multiple that both denominators have in common.
• Combine like terms: Combine like terms on both sides of the equation to simplify the equation.
• Add or subtract the same value to both sides: Add or subtract the same value to both sides of the equation to isolate the variable.

Practice Problems

Here are some practice problems to help you practice solving rational equations:

1. Solve the equation: $\frac{x}{2} + \frac{3}{4} = \frac{5}{6}$
2. Solve the equation: $\frac{2x}{3} - \frac{4}{5} = \frac{3}{4}$
3. Solve the equation: $\frac{x}{2} + \frac{2}{3} = \frac{5}{6}$

Answer Key

Here are the answers to the practice problems:

1. $x = \frac{13}{6}$
2. $x = \frac{17}{10}$
3. $x = \frac{7}{6}$