7.4 Solve Rational Equations

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