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Understanding Rational Expressions: Evaluation, Restriction, Simplification, and Equivalence

This chapter covers the fundamental properties of rational expressions, including finding numerical values, determining restrictions, writing expressions in lowest terms, and recognizing equivalent forms.

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Understanding Rational Expressions: Evaluation, Restriction, Simplification, and Equivalence

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  1. Chapter 6 Section 1

  2. The Fundamental Property of Rational Expressions Find the numerical value of a rational expression. Find the values of the variable for which a rational expression is undefined. Write rational expressions in lowest terms. Recognize equivalent forms of rational expressions. 6.1 2 3 4

  3. Rational Expressions Examples of rational expressions: Rational expressions cannot have a denominator equal to 0 Slide 6.1-3

  4. Objective 1 Find the numerical value of a rational expression. Slide 6.1-4

  5. Find the value of the rational expression, when x = 3. CLASSROOM EXAMPLE 1 Evaluating Rational Expressions Solution: Slide 6.1-5

  6. Objective 2 Find the values of the variable for which a rational expression is undefined. Slide 6.1-6

  7. The 11th Commandment For instance, in the rational expression • If xis 2, then the denominator becomes 0, making the expression undefined. • Thus, xcannot equal 2. We indicate this restriction by writing x ≠ 2. Denominator cannot equal 0 Thou shall not… divide by zero The denominator of a rational expression cannot equal 0 because Division by 0 is Undefined Slide 6.1-7

  8. Finding Restrictions on the Variable Step 2:Solve this equation. Step 3:The solutions of the equation are the values that make the rational expression undefined. The variable cannot equal these values. Step 1:Set the denominator of the rational expression equal to 0. Determining When a Rational Expression is Undefined Slide 6.1-8

  9. Find any values of the variable for which each rational expression is undefined. CLASSROOM EXAMPLE 2 Finding Values That Make Rational Expressions Undefined Solution: never undefined Slide 6.1-9

  10. Objective 3 Write rational expressions in lowest terms. Slide 6.1-10

  11. Write rational expressions in lowest terms. Fundamental Property of Rational Expressions where K ≠ 0 and Q≠ 0 This property is based on the identity property of multiplication, since Lowest Terms If the greatest common factor of its numerator and denominator is 1. Slide 6.1-11

  12. Write each rational expression in lowest terms. CLASSROOM EXAMPLE 3 Writing in Lowest Terms Solution: Slide 6.1-12

  13. Step 1:Factorthe numerator and denominator completely. Addends cannot be divided out. Writing a Rational Expression in Lowest Terms Step 2:Use the fundamental propertyto divide out any common factors. Only common factors can be divided out, not common addends!!! Like This NOT like this! Slide 6.1-13

  14. Write in lowest terms. CLASSROOM EXAMPLE 4 Writing in Lowest Terms Solution: Slide 6.1-14

  15. Quotient of Opposites If the numerator and the denominator of a rational expression are opposites, as in then the rational expression is equal to −1. CLASSROOM EXAMPLE 5 Writing in Lowest Terms (Factors Are Opposites) Write in lowest terms. Slide 6.1-15

  16. Write each rational expression in lowest terms. CLASSROOM EXAMPLE 6 Writing in Lowest Terms (Factors Are Opposites) Solution: or Slide 6.1-16

  17. Objective 4 Recognize equivalent forms of rational expressions. Slide 6.1-17

  18. Recognize equivalent forms of rational expressions Three ways to write the common fraction = = The − sign representing the factor −1 is in front of the expression. = = The factor may instead be placed in the numeratoror in the denominator. Slide 6.1-18

  19. Choose the equivalent expression. Explain. a. b. Be careful to apply the distributive property correctly. Recognize equivalent forms of rational expressions Slide 6.1-19

  20. Write four equivalent forms of the rational expression. CLASSROOM EXAMPLE 7 Writing Equivalent Forms of a Rational Expression Solution: Slide 6.1-20

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