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Solving Equations Containing Rational Expressions
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  1. Solving Equations Containing Rational Expressions Unit 4 Lesson 9.6 text book CCSS: A.CED.1

  2. Standards for Mathematical Practice • 1. Make sense of problems and persevere in solving them. • 2.Reason abstractly and quantitatively. • 3. Construct viable arguments and critique the reasoning of others. • 4. Model with mathematics. • 5. Use appropriate tools strategically. • 6.Attend to precision. • 7. Look for and make use of structure. • 8.Look for and express regularity in repeated reasoning.

  3. CCSS: A.CED.1 • Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

  4. Essential Question(s): • How do I solve a rational equation? •  How do I use rational equations to solve problems?

  5. Recap of This Unit • So far in this unit we have: • Talked about Polynomial Functions with one variable. • Graphed polynomial functions with one variable. • Learned how to use Quadratic Techniques. • Talked about the Reminder and Factor Theorem. • Roots and Zeros.

  6. NEW this section Next up… • In this section, we will apply our knowledge of solving polynomial equations to solving rational equations and inequalities. Please try to control your excitement.

  7. Example 1 • Solve the equation below: • When each side of the equation is a single rational expression, we can use cross multiplication. • It is VERY important to check your answer in the original equation.

  8. Example 2 • Solve the equations below by cross-multiplying. Check your solution(s).

  9. LCDs: Performance-Enhancing Math Term? • When a rational equation is not expressed as a proportion (with one term on each side), we can solve it by multiplying each side of the equation by the least common denominator (LCD) of the rational equation. • NOTE: To balance the equation, we must be sure to multiply by the same quantity on both sides of the equation. 7-time Cy Young Award winner Roger Clemens never “knowingly used LCDs” in his career.

  10. “One less thing to worry about” If your solution doesn’t work in the original equation, well, I guess that’s just one less thing you has to worry about. • If one or more of your solutions are not valid in the original equation, they are called “extraneous solutions” and should not be included in your list of actual solutions to the equation. • The graphs of rational functions may have breaks in continuity. Breaks in continuity may appear as asymptote (a line that the graph of the function approaches, but never crosses) or as a point of discontinuity.

  11. Solving rational equations • Find the LCM for the denominators • Any solution that results in a zero in denominator must be excluded from your list of solutions. • Multiply both sides of the equation by the LCM to get rid of all denominators • Solve the resulting equation (may need quadratic techniques, etc.) • Always check your answers by substituting back into the original equation! WHY????

  12. Example 3 • Solve the equation by using the LCD. Check for extraneous solutions.

  13. Example 4 • Solve the equation by using the LCD. Check for extraneous solutions.

  14. Example Real life 5 • Suppose the population density in the Wichita, Kansas, area is related to the distance from the center of the city. • This is modeled by where D is the population density (in people per square mile) and x is the distance (in miles) from the center of the city. • Find the distance(s) where the population density is 375 people per square mile.

  15. INEQUALITIES • Recall that for inequalities, we often pretend we are dealing with an equation, put the solutions on a number line, and then test a point from each region • Same thing here! • 1st find the excluded values • Then solve the related equation • Put the solutions and excluded values on a number line • Then test a point in each region to determine which range(s) of values represent solutions!

  16. Solve the Inequalities Remember to be careful with multiply (negative number changes the direction of the inequality)

  17. Solve the Inequalities Multiply by the LCM which is 9x.

  18. Solve the Inequalities Was there any excluded values?

  19. Solve the Inequalities Was there any excluded values? YES

  20. Solve the Inequalities Using the exclude value and the solution Make a Number line. Test the values between the dotted lines.

  21. Solve the Inequalities Using the exclude value and the solution Make a Number line. Test the values between the dotted lines.

  22. Solve the Inequalities Using the exclude value and the solution Make a Number line. Test the values between the dotted lines.

  23. Solve the Inequalities Using the exclude value and the solution Make a Number line. Test the values between the dotted lines.

  24. So final solution will be:

  25. Real Life example: The function models the population, in thousands, of Nickelford, t years after 1997. The population, in thousands, of nearby New Ironfield is modeled by Determine the time period when the population of New Ironfield exceeded the population of Nickelford.

  26. Continue: **Solve for the interval(s) of t where Q(t) > P(t)

  27. Solving the Inequality Graphically • Graph both curves on the same set of axes. • Find the POIs of the two curves. • Use the POI to determine the intervals of t that • satisfy the inequality.

  28. Solving Rational Equations & Inequalities Practice