Rational equations are algebraic expressions with fractions. They're tricky because you need to find common denominators and watch out for sneaky extraneous solutions. But don't worry, we've got a game plan!

We'll learn how to solve these equations step-by-step, from finding common denominators to checking our answers. We'll also see how rational equations pop up in real-life problems and practice isolating variables. It's all about breaking it down and taking it slow.

Solving Rational Equations

Solving rational equations

• Find the least common denominator (LCD) of all rational terms in the equation
  • Factor each denominator into prime factors (2, 3, 5, 7)
  • The LCD is the product of all factors, using the highest power of each factor present in any denominator
• Multiply both sides of the equation by the LCD to clear the denominators
  • Distribute the LCD to each term on both sides of the equation, eliminating all denominators
• Simplify the resulting equation by combining like terms
• Solve the simplified equation using appropriate methods
  • Factor the equation if possible (quadratic equations)
  • Apply the quadratic formula if factoring is not possible ($ax^2 + bx + c = 0$)
• Check for extraneous solutions by substituting the solutions back into the original equation
  • Extraneous solutions occur when a solution causes a denominator to equal zero, resulting in an undefined expression
  • Discard any extraneous solutions, as they are not valid for the original rational equation

Applications of rational functions

• Identify the given information and the unknown quantity in the problem
• Set up a rational equation that models the problem situation
  • Assign variables to represent unknown quantities (let $x$ represent the number of items)
  • Express relationships between quantities using rational expressions (total cost = fixed cost + variable cost)
• Solve the rational equation using the methods described in solving rational equations
• Interpret the solution in the context of the original problem
  • Ensure that the solution makes sense given the problem context (positive number of items, reasonable cost)
  • Consider any limitations or constraints imposed by the problem situation (maximum budget, minimum order quantity)

Variable isolation in rational equations

• Identify the variable to be isolated in the rational equation
• Perform algebraic operations to isolate the desired variable on one side of the equation
  • Multiply both sides of the equation by the LCD to clear the denominators
  • Use the distributive property to simplify the equation ($a(b+c) = ab + ac$)
  • Add, subtract, multiply, or divide both sides of the equation by the same value to isolate the variable
• Solve the resulting equation for the desired variable
  • Factor the equation if necessary ($x^2 - 4 = (x+2)(x-2)$)
  • Apply the quadratic formula if the equation is quadratic ($x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$)
• Check the solution by substituting it back into the original equation
  • Verify that the solution does not result in a denominator equal to zero, which would indicate an extraneous solution

Understanding Rational Expressions

• Rational expressions are algebraic fractions that represent the quotient of two polynomials
• Simplify rational expressions by factoring both the numerator and denominator
• Identify common factors between the numerator and denominator to cancel out
• Recognize that a rational expression with a reciprocal can be rewritten by flipping the fraction and changing the operation
• Use factoring techniques to simplify complex rational expressions involving polynomials