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COLLEGE ALGEBRA

Learn how to solve rational equations and work rate problems in algebra. Understand the steps and restrictions involved in finding the solutions.

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COLLEGE ALGEBRA

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  1. COLLEGE ALGEBRA LIAL HORNSBY SCHNEIDER

  2. 1.6 Other Types of Equations and Applications Rational Expressions Work Rate Problems Equations with Radicals Equations Quadratic in Form

  3. Rational Equations A rational equation is an equation that has a rational expression for one or more terms. Because a rational expression is not defined when its denominator is 0, values of the variable for which any denominator equals 0 cannot be solutions of the equation. To solve a rational equation, begin by multiplying both sides by the least common denominator of the terms of the equation.

  4. SOLVING RATIONAL EQUATIONS THAT LEAD TO LINEAR EQUATIONS Example 1 Solve each equation. a. Solution The least common denominator is 3(x – 1), which is equal to 0 if x = 1. Therefore, 1 cannot possibly be a solution of this equation.

  5. SOLVING RATIONAL EQUATIONS THAT LEAD TO LINEAR EQUATIONS Example 1 Solve each equation. a. Solution Multiply by the LCD, 3(x – 1), where x ≠ 1.

  6. SOLVING RATIONAL EQUATIONS THAT LEAD TO LINEAR EQUATIONS Example 1 Solve each equation. a. Solution Simplify on both sides. Multiply.

  7. SOLVING RATIONAL EQUATIONS THAT LEAD TO LINEAR EQUATIONS Example 1 Solve each equation. a. Solution Multiply. Subtract 3x2; combine terms. Solve the linear equation. The restriction x ≠ 1 does not affect this result.

  8. SOLVING RATIONAL EQUATIONS THAT LEAD TO LINEAR EQUATIONS Example 1 Solve each equation. a. Solution The restriction x ≠ 1 does not affect this result.

  9. SOLVING RATIONAL EQUATIONS THAT LEAD TO LINEAR EQUATIONS Example 1 Solve each equation. b. Solution Multiply by the LCD, x – 2, where x ≠ 2.

  10. SOLVING RATIONAL EQUATIONS THAT LEAD TO LINEAR EQUATIONS Example 1 Solve each equation. b. Solution Distributive property Subtract 2x; combine like terms. Multiply by –1.

  11. SOLVING RATIONAL EQUATIONS THAT LEAD TO LINEAR EQUATIONS Example 1 Solve each equation. b. Solution The only proposed solution is 2. However, the variable is restricted to real numbers except 2; if x = 2, then multiplying by x – 2 in the first step is multiplying both sides by 0, which is not valid. Thus, the solution set is .

  12. SOLVING RATIONAL EQUATIONS THAT LEAD TO QUADRATIC EQUATIONS Example 2 Solve each equation. a. Solution Factor the last denominator. Multiply by x(x–2), x ≠ 0, 2.

  13. SOLVING RATIONAL EQUATIONS THAT LEAD TO QUADRATIC EQUATIONS Example 2 Solve each equation. a. Solution Multiply by x(x–2), x ≠ 0, 2. Distributive property

  14. SOLVING RATIONAL EQUATIONS THAT LEAD TO QUADRATIC EQUATIONS Example 2 Solve each equation. a. Solution Distributive property Standard form Factor.

  15. SOLVING RATIONAL EQUATIONS THAT LEAD TO QUADRATIC EQUATIONS Example 2 Solve each equation. a. Solution Factor. Zero-factor property Set each factor equal to 0. or Proposed solutions or

  16. SOLVING RATIONAL EQUATIONS THAT LEAD TO QUADRATIC EQUATIONS Example 2 Solve each equation. a. Solution Because of the restriction x ≠ 0, the only valid solution is–1. The solution set is {–1}.

  17. SOLVING RATIONAL EQUATIONS THAT LEAD TO QUADRATIC EQUATIONS Example 2 Solve each equation. b. Solution Factor.

  18. SOLVING RATIONAL EQUATIONS THAT LEAD TO QUADRATIC EQUATIONS Example 2 Solve each equation. b. Solution

  19. SOLVING RATIONAL EQUATIONS THAT LEAD TO QUADRATIC EQUATIONS Example 2 Solve each equation. b. Solution Distributive property Standard form Divide by –4.

  20. SOLVING RATIONAL EQUATIONS THAT LEAD TO QUADRATIC EQUATIONS Example 2 Solve each equation. b. Solution Divide by –4. Factor. Zero-factor property or Proposed solutions or

  21. SOLVING RATIONAL EQUATIONS THAT LEAD TO QUADRATIC EQUATIONS Example 2 Solve each equation. b. Solution Proposed solutions or Since the restrictions on x are x ≠ 1, –1, neither proposed solution is valid, so the solution set is .

  22. Work Rate Problems Problem Solving If a job can be done in t units of time, then the rate of work is 1/t of the job per time unit. Therefore, rate  time = portion of the job completed. If the letters r, t, and A represent the rate at which work is done, the time, and the amount of work accomplished, respectively, then

  23. Work Rate Problems Problem Solving Amounts of work are often measured in terms of the number of jobs accomplished. For instance, if one job is accomplished in t time units, then A = 1 and

  24. SOLVING A WORK RATE PROBLEM Example 3 One computer can do a job twice as fast as another. Working together, both computers can do the job in 2 hr. How long would it take each computer, working alone, to do the job?

  25. SOLVING A WORK RATE PROBLEM Example 3 Solution Step 1 Read the problem. We must find the time it would take each computer working alone to do the job.

  26. SOLVING A WORK RATE PROBLEM Example 3 Solution Step 2 Assign a variable. Let x represent the number of hours it would take the faster computer, working alone, to do the job. The time for the slower computer to do the job alone is then 2x. Therefore, and

  27. SOLVING A WORK RATE PROBLEM Example 3 Solution Step 2 Assign a variable. A = rt

  28. SOLVING A WORK RATE PROBLEM Example 3 Solution Step 3 Write an equation. The sum of the two parts of the job together is 1, since one whole job is done. Part of the job done by the faster computer Part of the job done by the slower computer One whole job + =

  29. SOLVING A WORK RATE PROBLEM Example 3 Solution Step 4 Solve. Multiply both sides by x. Distributive property Multiply. Add.

  30. SOLVING A WORK RATE PROBLEM Example 3 Solution Step 5 State the answer. The faster computer would take 3 hr to do the job alone, while the slower computer would take 2(3) = 6 hr. Be sure to give both answers here. Step 6 Check. The answer is reasonable, since the time working together (2 hr) is less than the time it would take the faster computer working alone.

  31. Rate Problem Note The sum of the rates of the individual computers is equal to their rate working together. Multiply both sides by 2x. Same solution found earlier

  32. Power Property If P and Q are algebraic expressions, then every solution of the equation P = Q is also a solution of the equation Pn = Qn , for any positive number.

  33. CautionBe very careful when using the power property. It does not say that the equations P = Q and Pn = Qn are equivalent; it only says that each solution of the original equation P = Q is also a solution of the new equation Pn = Qn.

  34. Solving an Equation Involving Radicals Step 1 Isolate the radical on one side of the equation. Step 2 Raise each side of the equation to a power that is the same as the index of the radical so that the radical is eliminated. If the equation still contains a radical, repeat Steps 1 and 2. Step 3 Solve the resulting equation. Step 4 Check each proposed solution in the original equation.

  35. SOLVING AN EQUATION CONTAINING A RADICAL Example 4 Solve Solution Isolate the radical. Square both sides.

  36. SOLVING AN EQUATION CONTAINING A RADICAL Example 4 Solve Solution: Solve the quadratic equation. Factor. or Zero-factor property or Proposed solutions

  37. SOLVING AN EQUATION CONTAINING A RADICAL Example 4 Solve Solution: or Proposed solutions Only 3 is a solution, giving the solution set {3}.

  38. SOLVING AN EQUATION CONTAINING TWO RADICALS Example 5 Solve Solution When an equation contains two radicals, begin by isolating one of the radicals on one side of the equation. Square both sides.

  39. SOLVING AN EQUATION CONTAINING TWO RADICALS Example 5 Solve Solution Square both sides. Be careful! Don’t forget this term when squaring. Isolate the remaining radical.

  40. SOLVING AN EQUATION CONTAINING TWO RADICALS Example 5 Solve Solution Isolate the remaining radical. Square again.

  41. SOLVING AN EQUATION CONTAINING TWO RADICALS Example 5 Solve Solution Solve the quadratic equation. or Proposed solutions or

  42. SOLVING AN EQUATION CONTAINING TWO RADICALS Example 5 Solve Solution Proposed solutions or Both 3 and –1 are solutions of the original equation, so {3, –1} is the solution set.

  43. CautionRemember to isolate a radical in Step 1. It would be incorrect to square each term individually as the first step in Example 5.

  44. SOLVING AN EQUATION CONTAINING A RADICAL (CUBE ROOT) Example 6 Solve Solution Isolate a radical. Cube both sides. Solve the quadratic equation.

  45. SOLVING AN EQUATION CONTAINING A RADICAL (CUBE ROOT) Example 6 Solve Solution Solve the quadratic equation. or Proposed solutions or

  46. SOLVING AN EQUATION CONTAINING A RADICAL (CUBE ROOT) Example 6 Solve Solution or Proposed solutions Both are valid solutions, and the solution set is {¼ ,1}.

  47. Equation Quadratic in Form An equation is said to be quadratic in form if it can be written as where a ≠ 0 and u is some algebraic expression.

  48. SOLVING EQUATIONS QUADRATIC IN FORM Example 7 a. Solve Solution Since Substitute. Factor. Zero-factor property or or

  49. SOLVING EQUATIONS QUADRATIC IN FORM Example 7 a. Solve Don’t forget this step. Zero-factor property or or Cube each side. or Proposed solutions or

  50. SOLVING EQUATIONS QUADRATIC IN FORM Example 7 a. Solve Solution or Proposed solutions The solution set is {–2, 7}.

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