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Splash Screen. Five-Minute Check (over Lesson 8–5) CCSS Then/Now New Vocabulary Example 1: Solve a Rational Equation Example 2: Solve a Rational Equation Example 3: Real-World Example: Mixture Problem Example 4: Real-World Example: Distance Problem

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 8–5) CCSS Then/Now New Vocabulary Example 1: Solve a Rational Equation Example 2: Solve a Rational Equation Example 3: Real-World Example: Mixture Problem Example 4: Real-World Example: Distance Problem Example 5: Real-World Example: Work Problems Key Concept: Solving Rational Inequalities Example 6: Solve a Rational Inequality Lesson Menu

  3. State whether represents a direct, joint, inverse, or combined variation. Then name the constant of variation. A.direct; B.joint; C.inverse; 2 D.combined; 2 5-Minute Check 1

  4. State whether represents a direct, joint, inverse, or combined variation. Then name the constant of variation. A.direct; B.joint; C.inverse; 2 D.combined; 2 5-Minute Check 1

  5. A.direct; 7.5 B.joint; 7.5 C.inverse; D.combined; 7.5 State whether 7.5x = y represents a direct, joint, inverse, or combined variation. Then name the constant of variation. 5-Minute Check 2

  6. A.direct; 7.5 B.joint; 7.5 C.inverse; D.combined; 7.5 State whether 7.5x = y represents a direct, joint, inverse, or combined variation. Then name the constant of variation. 5-Minute Check 2

  7. If y varies inversely as x and y = 8 when x = 12, find y when x = 15. A. 4.8 B. 6.4 C. 8.6 D. 10.2 5-Minute Check 3

  8. If y varies inversely as x and y = 8 when x = 12, find y when x = 15. A. 4.8 B. 6.4 C. 8.6 D. 10.2 5-Minute Check 3

  9. If y varies jointly as x and z and y = 45 when x = 10 and z = 3, find y when x = 2 and z = 5. A. 9 B. 11 C. 13 D. 15 5-Minute Check 4

  10. If y varies jointly as x and z and y = 45 when x = 10 and z = 3, find y when x = 2 and z = 5. A. 9 B. 11 C. 13 D. 15 5-Minute Check 4

  11. A map shows the scale 1.5 inches equals 65 miles. How many miles apart are two cities if they are 7.5 inches apart on the map? A. 487.5 mi B. 357.5 mi C. 325 mi D. 260 mi 5-Minute Check 5

  12. A map shows the scale 1.5 inches equals 65 miles. How many miles apart are two cities if they are 7.5 inches apart on the map? A. 487.5 mi B. 357.5 mi C. 325 mi D. 260 mi 5-Minute Check 5

  13. The amount of interest earned on a savings account varies jointly with time and the amount deposited. After 5 years, interest on $1000 in the savings account is $225. What is the annual interest rate (constant of variation)? A. 2% B. 3.5% C. 4% D. 4.5% 5-Minute Check 6

  14. The amount of interest earned on a savings account varies jointly with time and the amount deposited. After 5 years, interest on $1000 in the savings account is $225. What is the annual interest rate (constant of variation)? A. 2% B. 3.5% C. 4% D. 4.5% 5-Minute Check 6

  15. Content Standards A.CED.1 Create equations and inequalities in one variable and use them to solve problems. A.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Mathematical Practices 6 Attend to precision. CCSS

  16. You simplified rational expressions. • Solve rational equations. • Solve rational inequalities. Then/Now

  17. rational equation • weighted average • rational inequality Vocabulary

  18. Solve . Check your solution. Solve a Rational Equation The LCD for the terms is 24(3 – x). Original equation Multiply each side by 24(3 – x). Example 1

  19. Solve a Rational Equation Distributive Property Simplify. Simplify. Add 6x and –63 to each side. Example 1

  20. Check Solve a Rational Equation Original equation x = –45 Simplify. Simplify. The solution is correct.  Example 1

  21. Solve a Rational Equation Answer: Example 1

  22. Solve a Rational Equation Answer: The solution is –45. Example 1

  23. Solve . A.–2 B. C. D.2 Example 1

  24. Solve . A.–2 B. C. D.2 Example 1

  25. Solve Check your solution. Solve a Rational Equation The LCD is (p + 1)(p – 1). Original equation Multiply by the LCD. Example 2

  26. Solve a Rational Equation (p – 1)(p2 – p – 5) = (p2 – 7)(p + 1) + p(p + 1)(p – 1) Divide common factors. p3 – p2 – 5p – p2 + p + 5 = p3 + p2 – 7p – 7 + p3 – p Distributive Property p3 – 2p2 – 4p + 5 = 2p3 + p2 – 8p – 7 Simplify. 0 = p3 + 3p2 – 4p – 12 Subtract p3 – 2p2 – 4p + 5 from each side. Example 2

  27. ? Solve a Rational Equation 0 = (p + 3)(p + 2)(p – 2) Factor. 0 = p + 3 or 0 = p + 2 or 0 = p – 2 Zero Product Property Check Try p = –3. Original equation p = –3 Example 2

  28. ? ? or Solve a Rational Equation Simplify. Simplify.  Try p = –2. Original equation Example 2

  29. ? ? ? Solve a Rational Equation p = –2 Simplify. Simplify.  Example 2

  30. ? ? ? Solve a Rational Equation Try p = 2. Original equation p = 2 Simplify. Simplify.  Answer: Example 2

  31. ? ? ? Solve a Rational Equation Try p = 2. Original equation p = 2 Simplify. Simplify.  Answer: The solutions are –3, –2 and 2. Example 2

  32. A. 4 B. –2 C. 2 D. –4 Example 2

  33. A. 4 B. –2 C. 2 D. –4 Example 2

  34. Mixture Problem BRINEAaron adds an 80% brine (salt and water) solution to 16 ounces of solution that is 10% brine. How much of the solution should be added to create a solution that is 50% brine? Understand Aaron needs to know how much of a solution needs to be added to an original solution to create a new solution. Example 3

  35. Percentage of brine in solution Mixture Problem Plan Each solution has a certain percentage that is salt. The percentage of brine in the final solution must equal the amount of brine divided by the total solution. Example 3

  36. Solve Write a proportion. Mixture Problem Substitute. Simplify numerator. LCD is 100(16 + x). Example 3

  37. Mixture Problem Divide common factors. Simplify. Distribute. Subtract 50x and 160. Divide each side by 30. Answer: Example 3

  38. Answer: Aaron needs to add ounces of 80% brine solution. Mixture Problem Divide common factors. Simplify. Distribute. Subtract 50x and 160. Divide each side by 30. Example 3

  39. Check Original equation ? ? Simplify. 0.5 = 0.5  Mixture Problem Simplify. Example 3

  40. BRINEJanna adds a 65% base solution to 13 ounces of solution that is 20% base. How much of the solution should be added to create a solution that is 40% base? A. 9.6 ounces B. 10.4 ounces C. 11.8 ounces D. 12.3 ounces Example 3

  41. BRINEJanna adds a 65% base solution to 13 ounces of solution that is 20% base. How much of the solution should be added to create a solution that is 40% base? A. 9.6 ounces B. 10.4 ounces C. 11.8 ounces D. 12.3 ounces Example 3

  42. Plan She swam 2 miles upstream against the current and 2 miles back to the dock with the current. The formula that relates distance, time, and rate is d = rt or Distance Problem SWIMMINGLilia swims for 5 hours in a stream that has a current of 1 mile per hour. She leaves her dock and swims upstream for 2 miles and then back to her dock. What is her swimming speed in still water? Understand We are given the speed of the current, the distance she swims upstream, and the total time. Example 4

  43. Time going withthe current plus time going againstthe current equals totaltime. 5 Original equation Distance Problem Let r equal her speed in still water. Then her speed with the current is r + 1, and her speed against the current is r – 1. Solve Example 4

  44. Divide Common Factors Multiply each side by r2 – 1. (r + 1)2 + (r – 1)2 = 5(r2 – 1) Simplify. Distribute. Simplify. Subtract 4r from each side. Distance Problem Example 4

  45. Simplify. Distance Problem Use the Quadratic Formula to solve for r. Quadratic Formula x = r, a = 5, b = – 4, and c = –5 Simplify. Example 4

  46. r≈ 1.5 or –0.7 Use a calculator. Distance Problem Answer: Example 4

  47. r≈ 1.5 or –0.7 Use a calculator. Check Original equation ? r = 1.5 ? Simplify. Simplify.  Distance Problem Answer: Since speed must be positive, the answer is about 1.5 miles per hour. Example 4

  48. SWIMMING Lynne swims for 1 hour in a stream that has a current of 2 miles per hour. She leaves her dock and swims upstream for 3 miles and then back to her dock. What is her swimming speed in still water? A. about 0.6 mph B. about 2.0 mph C. about 4.6 mph D. about 6.6 mph Example 4

  49. SWIMMING Lynne swims for 1 hour in a stream that has a current of 2 miles per hour. She leaves her dock and swims upstream for 3 miles and then back to her dock. What is her swimming speed in still water? A. about 0.6 mph B. about 2.0 mph C. about 4.6 mph D. about 6.6 mph Example 4

  50. Plan Wuyi can mow the lawn in 4.5 hours, so the rate of mowing is of a lawn per hour. Work Problems MOWING LAWNSWuyi and Uima mow lawns together. Wuyi working alone could complete a particular job in 4.5 hours, and Uima could complete it alone in 3.7 hours. How long does it take to complete the job when they work together? Understand We are given how long it takes Wuyi and Uima working alone to mow a particular lawn. We need to determine how long it would take them together. Example 5

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