U2 L2.1: Solving Rational Equations

• The latter 2 examples are not considered as rational expressions since they do not show a ratio between two polynomials
• Rational Expression


  A ratio between two polynomials, like a fraction showing the relationship of one quantity to another
• Classify ratios as rational expressions or not

  • 8/(7x), 2. (5-6x)/3, 3. (4+3x)/(x-1), 4. (8x^3-1)/(4x^2+4x+1), 5. 2x/(x+2)
• Solving Rational Equations is an eq
• Topics to revisit before starting the lesson

  • Fractions and Rational Expressions
  • Operations on Fractions
  • Prime Factorization
  • Finding the Least Common Denominator
  • Solving Linear and Quadratic Equations
• For a polynomial: The variable cannot be inside a radical sign. The variable should not have a fractional or a negative exponent
• Rational Expressions

  • 3/(2x)/(7x-1)
  • (x^2-5)/(x^3+8)
  • (4x^2-4x+1)/(x^3+2x^2-3x+6)/(x+3)
• Examples of Rational Equations

  • 2/5y = 4, 10x/(x-3) = x + 1/-4, 2k/(k-4) - 2 = 4/k
• Simplified Rational Expression

  • 8(x+1)/(x-2) or 8x+8/(x-2)
• A rational equation is a type of equation involving at least one rational expression
• All numerators and denominators in the given ratios are considered polynomials
• Mathematical Statements using Rational Expressions


  Rational Equation - an equality involving at least 1 rational expression, Rational Inequality - an inequality involving at least 1 rational expression, Rational Function - a function that involves a rational expression
• Denominators


  • 𝑥2 − 2𝑥, 𝑥, 𝑥 − 2
  • 10𝑟 − 5, 2𝑟 − 1
  • 7𝑡, 𝑡2, 𝑡2 + 3𝑡
• Solving Rational Equations

  1. As an equation, it tells that both sides and quantities have the same value

  2. To solve for a rational equation means finding a value or a set of values that will satisfy the equation

  3. In eliminating the denominator from a rational equation, the least common multiple of the denominators (or the least common denominator or LCD) will be used. This must be multiplied to each term of the equation, cancelling out common factors that will be observed from the numerator and denominator

  4. Determining the least common multiple makes use of the prime factors of the denominators

  5. List down the denominators and their corresponding prime factors. Align the common factors that may be observed. Afterwards, get the factor from each column to be the factors of the LCD

  6. Hence, in the equation, the least common denominator to be used is or its factored form

  7. Now, let these be the denominators from a given rational equation
• What would be the least common multiple between these polynomials
• Rational Equation is a type of equation involving at least one rational expression
• To solve for a rational equation means finding a value or a set of values that will satisfy the equation
• What would be the least common multiple between these polynomials that will serve as the LCD in solving the equation?
• When checking the solution if it satisfies the given equation, solve both sides of the equation separately. Never make a transposition
• If the equation results in equal values on both sides, then it is considered as a true solution. In case it makes the denominator zero making the rational expression undefined, it is not a true solution and is called an extraneous solution
• Least common multiple of the following polynomials

  • 𝑏2, 4𝑏, �� − 1
  • 𝑥2 + 3𝑥, 𝑥2 − 9
  • 𝑥2 − 4, ��2 − 4𝑥 + 4
• How to solve rational equations?

  1. Determine the least common denominator (LCD) or the least common multiple of the denominators

  2. Multiply the entire equation by the LCD to eliminate the denominator

  3. Solve the remaining linear, quadratic or polynomial equation to find the solution for the unknown

  4. Check the solution if it satisfies the given equation by substituting it to the unknown
• Solution for Example 1

  1. Determine the least common denominator (LCD) or the least common multiple of the denominators

  2. Multiply the entire equation by the LCD to eliminate the denominator

  3. Solve the remaining linear, quadratic or polynomial equation to find the solution for the unknown

  4. Check the solution if it satisfies the given equation by substituting it to the unknown
• Example 1: Solve for y in 2/5�� = 4/10
• The possible values of x are 1 and -3
• Solving Rational Equations

  Determine the least common denominator (LCD) or the least common multiple of the denominators
• Solving Rational Equations

  Solve for x in 𝑥/(𝑥 − 3) = 𝑥 + 1/−4
• Solving Rational Equations

  Solve for y in 𝑦/(𝑦 − 2) + 3/(𝑦 + 2) = 𝑦 − 1/(𝑦^2 − 4)
• Both -3 and 1 are true solutions for the given rational equation 𝑥/(𝑥 − 3) = 𝑥 + 1/−4
• Solving Rational Equations

  Determine the least common denominator (LCD)
• Solving Rational Equations


  Multiply the entire equation by the LCD to eliminate the denominator
• 1 is a true solution to the equation since the resulting values on both sides of the equation are equal
• Solving Rational Equations


  Solve the remaining linear, quadratic or polynomial equation to find the solution for the unknown
• The possible value of y is 1
• Solving Rational Equations


  Check the solution if it satisfies the given equation by substituting it to the unknown
• Let y = 1 in the given equation. Hence, 1 satisfies the equation
• Example 4

  Solve for x in 𝑥/(𝑥 + 3) + (𝑥 − 1)/(𝑥 − 3) = (�� + 9)/(��2 − 9)
• The possible value of y are −5 and 1
• Example 5

  Solve for r in 10r(r − 2) − 4/r = 5/(r − 2)
• The possible value of x are −2 and 3