Section 6.4 Rational Equations Solving Rational Equations Clearing Fractions in an Equation Restricted Domains (and Solutions) The Principle of Zero Products The Necessity of Checking Rational Equations and Graphs 6.4

A Rational Equation in One Variable May Have Solution(s) A Rational Equation contains at least one Rational Expression. Examples: All Solution(s) must be tested in the Original Equation 6.4

False Solutions Warning: Clearing an equation may add a False Solution A False Solution is one that causes an untrue equation, or a divide by zero situation in the original equation Before even starting to solve a rational equation, we need to identify values to be excluded What values need to be excluded for these? t ≠ 0 a ≠ ±5 x ≠ 0 6.4

Clearing Factions from Equations Review: Simplify - Clear a Complex Fraction by Multiplying top and bottom by the LCD Solve - Clear a Rational Equation by Multiplying both sides by the LCD Then solve the new polynomial equation using the principle of zero products 6.4

The Principle of Zero Products Covered in more detail in Section 5.8 When a polynomial equation is in form polynomial = 0 you can set each factor to zero to find solution(s) Example – What are the solutions to: x2 – x – 6 = 0 (x – 3)(x + 2) = 0 x – 3 = 0  x = 3 and x + 2 = 0  x = -2 6.4

Clearing & Solving a Rational Equation What gets excluded? x ≠ 0 What’s the LCD? 15x What’s the solution? 6.4

A Binomial Denominator What gets excluded? x ≠ 5 What’s the LCD? x – 5 What’s the solution? 6.4

Another Binomial Denominator What gets excluded? x ≠ 3 What’s the LCD? x – 3 What’s the solution? x = -3 (x = 3 excluded) 6.4

Different Binomial Denominators What gets excluded? x ≠ 5,-5 What’s the LCD? (x – 5)(x + 5) What’s the solution? x = 7 6.4

Functions as Rational Equations What gets excluded? x ≠ 0 What’s the LCD? x What’s the solution? x = 2 and x = 3 6.4

What Next? 6.5 Solving Applications of Rational Equations 6.4