Solve The Rational Equation. Express Numbers As Integers Or Simplified Fractions.${ \frac{x}{x-8} + \frac{1}{8} = \frac{2}{x-8} }$The Solution Set Is { \square$}$.

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Introduction

Rational equations are a fundamental concept in algebra, and solving them requires a deep understanding of fractions, algebraic manipulation, and problem-solving strategies. In this article, we will focus on solving a specific rational equation, xx−8+18=2x−8\frac{x}{x-8} + \frac{1}{8} = \frac{2}{x-8}, and provide a step-by-step guide on how to express numbers as integers or simplified fractions.

Understanding Rational Equations

A rational equation is an equation that contains one or more rational expressions, which are fractions that contain variables in the numerator or denominator. Rational equations can be solved using various techniques, including factoring, cross-multiplication, and algebraic manipulation.

Key Concepts

  • Rational Expressions: A rational expression is a fraction that contains variables in the numerator or denominator.
  • Simplifying Rational Expressions: Simplifying rational expressions involves canceling out common factors in the numerator and denominator.
  • Cross-Multiplication: Cross-multiplication is a technique used to solve rational equations by multiplying both sides of the equation by the least common multiple (LCM) of the denominators.

Solving the Rational Equation

To solve the rational equation xx−8+18=2x−8\frac{x}{x-8} + \frac{1}{8} = \frac{2}{x-8}, we will follow a step-by-step approach.

Step 1: Multiply Both Sides by the LCM

The least common multiple (LCM) of the denominators is (x−8)×8=8(x−8)(x-8) \times 8 = 8(x-8). We will multiply both sides of the equation by the LCM to eliminate the fractions.

xx−8×8(x−8)+18×8(x−8)=2x−8×8(x−8)\frac{x}{x-8} \times 8(x-8) + \frac{1}{8} \times 8(x-8) = \frac{2}{x-8} \times 8(x-8)

This simplifies to:

8x+(x−8)=168x + (x-8) = 16

Step 2: Simplify the Equation

We will simplify the equation by combining like terms.

8x+x−8=168x + x - 8 = 16

This simplifies to:

9x−8=169x - 8 = 16

Step 3: Add 8 to Both Sides

We will add 8 to both sides of the equation to isolate the term with the variable.

9x−8+8=16+89x - 8 + 8 = 16 + 8

This simplifies to:

9x=249x = 24

Step 4: Divide Both Sides by 9

We will divide both sides of the equation by 9 to solve for x.

9x9=249\frac{9x}{9} = \frac{24}{9}

This simplifies to:

x=249x = \frac{24}{9}

Step 5: Simplify the Fraction

We will simplify the fraction by dividing both the numerator and denominator by their greatest common divisor (GCD).

x=249=83x = \frac{24}{9} = \frac{8}{3}

Conclusion

Solving rational equations requires a deep understanding of fractions, algebraic manipulation, and problem-solving strategies. By following a step-by-step approach, we can solve rational equations and express numbers as integers or simplified fractions. In this article, we solved the rational equation xx−8+18=2x−8\frac{x}{x-8} + \frac{1}{8} = \frac{2}{x-8} and found the solution set to be x=83x = \frac{8}{3}.

Frequently Asked Questions

Q: What is a rational equation?

A: A rational equation is an equation that contains one or more rational expressions, which are fractions that contain variables in the numerator or denominator.

Q: How do I solve a rational equation?

A: To solve a rational equation, you can follow a step-by-step approach, including multiplying both sides by the LCM, simplifying the equation, adding or subtracting terms, and dividing both sides by the coefficient of the variable.

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

A: The least common multiple (LCM) is the smallest multiple that two or more numbers have in common. In the context of rational equations, the LCM is used to eliminate fractions by multiplying both sides of the equation by the LCM.

Additional Resources

  • Khan Academy: Rational Equations
  • Mathway: Rational Equations
  • Wolfram Alpha: Rational Equations

Final Thoughts

Solving rational equations requires a deep understanding of fractions, algebraic manipulation, and problem-solving strategies. By following a step-by-step approach, we can solve rational equations and express numbers as integers or simplified fractions. In this article, we solved the rational equation xx−8+18=2x−8\frac{x}{x-8} + \frac{1}{8} = \frac{2}{x-8} and found the solution set to be x=83x = \frac{8}{3}. We hope this article has provided valuable insights and techniques for solving rational equations.

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Introduction

Rational equations are a fundamental concept in algebra, and solving them requires a deep understanding of fractions, algebraic manipulation, and problem-solving strategies. In this article, we will provide a comprehensive Q&A section on rational equations, covering frequently asked questions and answers.

Q&A Section

Q: What is a rational equation?

A: A rational equation is an equation that contains one or more rational expressions, which are fractions that contain variables in the numerator or denominator.

Q: How do I solve a rational equation?

A: To solve a rational equation, you can follow a step-by-step approach, including multiplying both sides by the LCM, simplifying the equation, adding or subtracting terms, and dividing both sides by the coefficient of the variable.

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

A: The least common multiple (LCM) is the smallest multiple that two or more numbers have in common. In the context of rational equations, the LCM is used to eliminate fractions by multiplying both sides of the equation by the LCM.

Q: How do I find the LCM of two numbers?

A: To find the LCM of two numbers, you can list the multiples of each number and find the smallest multiple that appears in both lists.

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

A: A rational expression is a fraction that contains variables in the numerator or denominator, while a rational equation is an equation that contains one or more rational expressions.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you can cancel out common factors in the numerator and denominator.

Q: What is the rule for multiplying rational expressions?

A: When multiplying rational expressions, you can multiply the numerators and denominators separately and then simplify the resulting expression.

Q: How do I solve a rational equation with multiple variables?

A: To solve a rational equation with multiple variables, you can use the same step-by-step approach as solving a rational equation with a single variable, but you may need to use additional techniques such as substitution or elimination.

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 rational expressions, while a quadratic equation is an equation that contains a quadratic expression, which is a polynomial of degree two.

Q: How do I solve a rational equation with a quadratic expression?

A: To solve a rational equation with a quadratic expression, you can use the same step-by-step approach as solving a rational equation, but you may need to use additional techniques such as factoring or the quadratic formula.

Additional Resources

  • Khan Academy: Rational Equations
  • Mathway: Rational Equations
  • Wolfram Alpha: Rational Equations

Final Thoughts

Solving rational equations requires a deep understanding of fractions, algebraic manipulation, and problem-solving strategies. By following a step-by-step approach and using additional techniques such as substitution or elimination, you can solve rational equations and express numbers as integers or simplified fractions. We hope this Q&A article has provided valuable insights and techniques for solving rational equations.

Common Mistakes to Avoid

  • Not simplifying the equation: Make sure to simplify the equation before solving it.
  • Not finding the LCM: Make sure to find the LCM of the denominators before multiplying both sides of the equation.
  • Not canceling out common factors: Make sure to cancel out common factors in the numerator and denominator.
  • Not using the correct technique: Make sure to use the correct technique for solving the rational equation, such as substitution or elimination.

Conclusion

Solving rational equations requires a deep understanding of fractions, algebraic manipulation, and problem-solving strategies. By following a step-by-step approach and using additional techniques such as substitution or elimination, you can solve rational equations and express numbers as integers or simplified fractions. We hope this Q&A article has provided valuable insights and techniques for solving rational equations.