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Algebra I, Chapter 11 Rational Expressions and Equations Standard (Chapter section): 1. 2. 3. Alg. 12.0 – Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms (11.2) Alg. 13.0 – Students add, subtract, multiply, and divide rational expressions and functions. Students solve both computationally and conceptually challenging problems by using these techniques (11.3-11.5) Alg. 15.0 – Students apply algebraic techniques to solve rate problems, work problems, an percent mixture problems (11.1, 11.6) Targets: 1. I can identify direct and inverse variation (11.1) 2. I can graph an inverse variation equation, both positive and negative (11.1) 3. I can match direct and inverse variation graphs with their equations (11.1) 4. I can tell whether or not a table represents inverse variation (11.1) 5. I can write an inverse variation equation from a data in a table (11.1) 6. I can find excluded values (11.2) 7. I can simplify rational expressions by dividing out monomials (11.2) 8. I can recognize opposites and use that information to simplify a rational equation (11.2) 9. I can multiply a rational expressions involving monomials (11.3) 10. I can multiply a rational expression by a polynomial (11.3) 11. I can divide a rational expression involving polynomials (11.3) 12. I can add or subtract rational expressions with common denominators (11.4) 13. I can find the LCD of two rational expressions (11.4) 14. I can add or subtract rational expressions with different denominators (11.4) 15. I can find the add or subtract two rational expressions by first factoring both expressions’ denominators in order to find the LCD (11.4) 16. I can use the Distance Traveled formula to solve a rational expression (11.4) 17. I can use cross-products to solve rational equations (11.5) 18. I can use properties of equality and the LCD to solve a rational equations (11.5) 19. I can factor to find the LCD and use properties of equality to solve a rational equation (11.5) 20. I can solve a Mixture problem involving rational expressions (11.5) 21. I can use the Work Rate formula to find the time it takes to complete a job (11.6) Vocabulary: (italics indicate a vocabulary word previously learned in an earlier chapter) Inverse variation Constant of variation Rational expression Excluded value Hyperbola Simplest form of a rational expression Rational function Multiplicative inverse Branches of a hyperbola Asymptotes of a hyperbola Polynomial Least common denominator (LCD) of rational expressions Rational equation Extraneous solution Cross product NOTES: Graphs of Direct Variation - y kx Graphs of Inverse Variation - y y y 2 2 1 –2 k x 1 –1 –1 1 x 2 –2 –1 –1 –2 1 x 2 –2 Positive (k>0): Positive (k>0): y y 2 2 1 –2 –1 –1 1 1 2 x –2 –2 –1 –1 1 2 x –2 Negative (k<0): Negative (k<0): Simplifying Rational Expressions By finding the GCF of the numerator and denominator and then dividing both the numerator and denominator by that GCF: 4 ( x 7) 2( x 3) 2 ( x 3) x3 6( x 2) 2 3( x 2) 3( x 2) 5 ( x 7) 4 5 By factoring both the numerator and denominator and then dividing both the numerator and denominator by the same binomial: 2 x 6 2 ( x 3) 2 3x 9 3 ( x 3) 3 Multiplying Rational Expressions Cross-cancel any common factors or Binomials Multiply the Numerators then multiply the denominators Check to see that it is completely simplified Adding & Subtracting Rational Expressions with LIKE denominators: Add numerators Use the LIKE denominator as the new denominator Simplify the rational expression by whatever method is appropriate Solve Rational Expressions with Cross-Products: 6 x x4 2 2(6) x( x 4) 12 x 2 4 x ( x 4) ( x 4) x 4 x 2 8 x 16 ( x 4) 2 x 2 16 ( x 4)( x 4) ( x 4) ( x 4) x 4 Dividing Rational Expressions Rewrite as a multiplication problem by: replacing the with taking the reciprocal of the divisor (2nd expression) Multiply Adding & Subtracting Rational Expressions with UNLIKE denominators: Find the LCD (least common denominator) Multiply the numerator & denominator of each rational expression to create expressions with LIKE denominators Then add or subtract as appropriate Work Problems: Work Rate Time = Work Done (usually 1 job) Work Rate = Work Done Time Bob takes three hours to mow one lawn. Bob’s Work Rate is 1 job per hour. 3 Sue takes 2 hours to mow one lawn. Sue’s Work Rate is 1 job per hour. 2 If they work together then: 0 x 2 4 x 12 0 ( x 6)( x 2) x 6 or x 2 t t 1 , multiply by LCD (6) 3 2 2t 3t 6 5t 6 t 6 5 Bob and Sue can mow one lawn together in 1 hour and 12 minutes.