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Chapter 5

Chapter 5 . 5-5 Solving Rational Equations and inequalities. Objectives. Solve rational equations and inequalities. Rational Equation.

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Chapter 5

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  1. Chapter 5 5-5 Solving Rational Equations and inequalities

  2. Objectives • Solve rational equations and inequalities.

  3. Rational Equation • A rational equation is an equation that contains one or more rational expressions. The time t in hours that it takes to travel d miles can be determined by using the equation t = , where r is the average rate of speed. This equation is a rational equation.

  4. Rational Equation • To solve a rational equation, start by multiplying each term of the equation by the least common denominator (LCD) of all of the expressions in the equation. This step eliminates the denominators of the rational expression and results in an equation you can solve by using algebra.

  5. Example • Solve the equation x – 18/ x = 3.

  6. Example • Solve the equation 10/3 = 4/x + 2.

  7. Example • Solve the equation 6/x + 5/4 = – 7/4 .

  8. Extraneous Solution • An extraneous solution is a solution of an equation derived from an original equation that is not a solution of the original equation. When you solve a rational equation, it is possible to get extraneous solutions. These values should be eliminated from the solution set. Always check your solutions by substituting them into the original equation.

  9. = 3x + 4 5x x – 2 x – 2 Example • Solve

  10. x 2 2x – 5 11 x – 8 x – 8 Example • Solve + =

  11. Solve the equation . = + x 6 1 x x – 1 x – 1 Example

  12. Problem solving application • A jet travels 3950 mi from Chicago, Illinois, to London, England, and 3950 mi on the return trip. The total flying time is 16.5 h. The return trip takes longer due to winds that generally blow from west to east. If the jet’s average speed with no wind is 485 mi/h, what is the average speed of the wind during the round-trip flight? Round to the nearest mile per hour.

  13. Application • On a river, a kayaker travels 2 mi upstream and 2 mi downstream in a total of 5 h. In still water, the kayaker can travel at an average speed of 2 mi/h. Based on this information, what is the average speed of the current of this river? Round to the nearest tenth.

  14. Rational inequality • A rational inequality is an inequality that contains one or more rational expressions. One way to solve rational inequalities is by using graphs and tables.

  15. Solve ≤ 3 by using a graph and a table. x x–6 Example

  16. Inequalities • You can also solve rational inequalities algebraically. You start by multiplying each term by the least common denominator (LCD) of all the expressions in the inequality. However, you must consider two cases: the LCD is positive or the LCD is negative.

  17. Solve ≤ 3 algebraically. 6 x–8 Example

  18. Solve ≥ –4 algebraically. 6 x–2 Example

  19. Student Guided Practice • Do even problems from 2-14 in your book page 353

  20. Homework • Do even problems 19-32 in your book page 353

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