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**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

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**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

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**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 ≠ a ≠ ± x ≠ 0 6.4

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**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

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**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

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**Clearing & Solving a Rational Equation**

What gets excluded? x ≠ 0 What’s the LCD? 15x What’s the solution? 6.4

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**A Binomial Denominator**

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

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**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

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**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

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**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

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What Next? 6.5 Solving Applications of Rational Equations 6.4

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