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• 1. General Rational Equations Frank Ma © 2011
• 2. General Rational Equations An equation or relation expressed using fractions may always be restated in a way without using fractions.
• 3. This is accomplished by multiplying the LCD of all the fractional terms in the equation, to each term on both sides, then clear each denominator using these multiplications. General Rational Equations An equation or relation expressed using fractions may always be restated in a way without using fractions.
• 4. Example A. Suppose the value of 3/4 of an apple is the same as the value of 1/2 of a banana with 1/3 of a cantaloupe, restate this relation in whole values. This is accomplished by multiplying the LCD of all the fractional terms in the equation, to each term on both sides, then clear each denominator using these multiplications. General Rational Equations An equation or relation expressed using fractions may always be restated in a way without using fractions.
• 5. Example A. Suppose the value of 3/4 of an apple is the same as the value of 1/2 of a banana with 1/3 of a cantaloupe, restate this relation in whole values. This is accomplished by multiplying the LCD of all the fractional terms in the equation, to each term on both sides, then clear each denominator using these multiplications. General Rational Equations An equation or relation expressed using fractions may always be restated in a way without using fractions. Let A = apple, B = banana, and C = cantaloupe
• 6. Example A. Suppose the value of 3/4 of an apple is the same as the value of 1/2 of a banana with 1/3 of a cantaloupe, restate this relation in whole values. This is accomplished by multiplying the LCD of all the fractional terms in the equation, to each term on both sides, then clear each denominator using these multiplications. General Rational Equations An equation or relation expressed using fractions may always be restated in a way without using fractions. Let A = apple, B = banana, and C = cantaloupe The relation is 3 4 A 1 2 B += 1 3 C
• 7. Example A. Suppose the value of 3/4 of an apple is the same as the value of 1/2 of a banana with 1/3 of a cantaloupe, restate this relation in whole values. This is accomplished by multiplying the LCD of all the fractional terms in the equation, to each term on both sides, then clear each denominator using these multiplications. General Rational Equations An equation or relation expressed using fractions may always be restated in a way without using fractions. The LCD =12, multiply it to both sides. 3 4 A 1 2 B += 1 3 C Let A = apple, B = banana, and C = cantaloupe The relation is
• 8. Example A. Suppose the value of 3/4 of an apple is the same as the value of 1/2 of a banana with 1/3 of a cantaloupe, restate this relation in whole values. This is accomplished by multiplying the LCD of all the fractional terms in the equation, to each term on both sides, then clear each denominator using these multiplications. General Rational Equations An equation or relation expressed using fractions may always be restated in a way without using fractions. The LCD =12, multiply it to both sides. 3 4 A 1 2 B += 3 4 A 1 2 B +=( ) 1 3 C 1 3 C * 12 Let A = apple, B = banana, and C = cantaloupe The relation is
• 9. Example A. Suppose the value of 3/4 of an apple is the same as the value of 1/2 of a banana with 1/3 of a cantaloupe, restate this relation in whole values. This is accomplished by multiplying the LCD of all the fractional terms in the equation, to each term on both sides, then clear each denominator using these multiplications. General Rational Equations An equation or relation expressed using fractions may always be restated in a way without using fractions. The LCD =12, multiply it to both sides. 3 4 A 1 2 B += 3 4 A 1 2 B +=( ) 1 3 C 1 3 C * 12 Distribute it to each term. Let A = apple, B = banana, and C = cantaloupe The relation is
• 10. Example A. Suppose the value of 3/4 of an apple is the same as the value of 1/2 of a banana with 1/3 of a cantaloupe, restate this relation in whole values. This is accomplished by multiplying the LCD of all the fractional terms in the equation, to each term on both sides, then clear each denominator using these multiplications. General Rational Equations An equation or relation expressed using fractions may always be restated in a way without using fractions. The LCD =12, multiply it to both sides. 3 4 A 1 2 B += 3 4 A 1 2 B +=( ) 1 3 C 1 3 C * 12 Distribute it to each term. 3 Let A = apple, B = banana, and C = cantaloupe The relation is
• 11. Example A. Suppose the value of 3/4 of an apple is the same as the value of 1/2 of a banana with 1/3 of a cantaloupe, restate this relation in whole values. This is accomplished by multiplying the LCD of all the fractional terms in the equation, to each term on both sides, then clear each denominator using these multiplications. General Rational Equations An equation or relation expressed using fractions may always be restated in a way without using fractions. The LCD =12, multiply it to both sides. 3 4 A 1 2 B += 3 4 A 1 2 B +=( ) 1 3 C 1 3 C * 12 Distribute it to each term. 3 6 4 Let A = apple, B = banana, and C = cantaloupe The relation is
• 12. Example A. Suppose the value of 3/4 of an apple is the same as the value of 1/2 of a banana with 1/3 of a cantaloupe, restate this relation in whole values. This is accomplished by multiplying the LCD of all the fractional terms in the equation, to each term on both sides, then clear each denominator using these multiplications. General Rational Equations An equation or relation expressed using fractions may always be restated in a way without using fractions. The LCD =12, multiply it to both sides. 3 4 A 1 2 B += 3 4 A 1 2 B +=( ) 1 3 C 1 3 C * 12 Distribute it to each term. 3 6 4 9A = 6B + 4C Let A = apple, B = banana, and C = cantaloupe The relation is
• 13. Example A. Suppose the value of 3/4 of an apple is the same as the value of 1/2 of a banana with 1/3 of a cantaloupe, restate this relation in whole values. This is accomplished by multiplying the LCD of all the fractional terms in the equation, to each term on both sides, then clear each denominator using these multiplications. General Rational Equations An equation or relation expressed using fractions may always be restated in a way without using fractions. The LCD =12, multiply it to both sides. 3 4 A 1 2 B += 3 4 A 1 2 B +=( ) 1 3 C 1 3 C * 12 Distribute it to each term. 3 6 4 9A = 6B + 4C Hence 9 apples is the same as 6 bananas with 4 cantaloupes. Let A = apple, B = banana, and C = cantaloupe The relation is
• 14. General Rational Equations To solve an equation with fractional terms, we first clear the fractions by multiplying both sides by the LCD.
• 15. General Rational Equations To solve an equation with fractional terms, we first clear the fractions by multiplying both sides by the LCD. Example B. Solve 3 x – 1 = 6 x + 2
• 16. General Rational Equations To solve an equation with fractional terms, we first clear the fractions by multiplying both sides by the LCD. Example B. Solve 3 x – 1 = 6 x + 2 Multiply both sides by the LCD = (x – 1)(x + 2)
• 17. General Rational Equations To solve an equation with fractional terms, we first clear the fractions by multiplying both sides by the LCD. Example B. Solve 3 x – 1 = 6 x + 2 Multiply both sides by the LCD = (x – 1)(x + 2) 3 x – 1 = 6 x + 2 ( ) * (x – 1)(x + 2)
• 18. General Rational Equations To solve an equation with fractional terms, we first clear the fractions by multiplying both sides by the LCD. Example B. Solve 3 x – 1 = 6 x + 2 Multiply both sides by the LCD = (x – 1)(x + 2) 3 x – 1 = 6 x + 2 ( ) * (x – 1)(x + 2) (x + 2)
• 19. General Rational Equations To solve an equation with fractional terms, we first clear the fractions by multiplying both sides by the LCD. Example B. Solve 3 x – 1 = 6 x + 2 Multiply both sides by the LCD = (x – 1)(x + 2) 3 x – 1 = 6 x + 2 ( ) * (x – 1)(x + 2) (x – 1)(x + 2)
• 20. 3(x + 2) = 6( x – 1) General Rational Equations To solve an equation with fractional terms, we first clear the fractions by multiplying both sides by the LCD. Example B. Solve 3 x – 1 = 6 x + 2 Multiply both sides by the LCD = (x – 1)(x + 2) 3 x – 1 = 6 x + 2 ( ) * (x – 1)(x + 2) (x – 1)(x + 2)
• 21. 3(x + 2) = 6( x – 1) 3x + 6 = 6x – 6 General Rational Equations To solve an equation with fractional terms, we first clear the fractions by multiplying both sides by the LCD. Example B. Solve 3 x – 1 = 6 x + 2 Multiply both sides by the LCD = (x – 1)(x + 2) 3 x – 1 = 6 x + 2 ( ) * (x – 1)(x + 2) (x – 1)(x + 2)
• 22. 3(x + 2) = 6( x – 1) 3x + 6 = 6x – 6 6 + 6 = 6x – 3x General Rational Equations To solve an equation with fractional terms, we first clear the fractions by multiplying both sides by the LCD. Example B. Solve 3 x – 1 = 6 x + 2 Multiply both sides by the LCD = (x – 1)(x + 2) 3 x – 1 = 6 x + 2 ( ) * (x – 1)(x + 2) (x – 1)(x + 2)
• 23. 3(x + 2) = 6( x – 1) 3x + 6 = 6x – 6 6 + 6 = 6x – 3x 12 = 3x 4 = x General Rational Equations To solve an equation with fractional terms, we first clear the fractions by multiplying both sides by the LCD. Example B. Solve 3 x – 1 = 6 x + 2 Multiply both sides by the LCD = (x – 1)(x + 2) 3 x – 1 = 6 x + 2 ( ) * (x – 1)(x + 2) (x – 1)(x + 2)
• 24. 3(x + 2) = 6( x – 1) 3x + 6 = 6x – 6 6 + 6 = 6x – 3x 12 = 3x 4 = x General Rational Equations To solve an equation with fractional terms, we first clear the fractions by multiplying both sides by the LCD. Example B. Solve 3 x – 1 = 6 x + 2 Multiply both sides by the LCD = (x – 1)(x + 2) 3 x – 1 = 6 x + 2 ( ) * (x – 1)(x + 2) (x – 1)(x + 2) Note that in this proportional equation, multiplying the LCD yield the same equation if we cross-multiplied.
• 25. General Rational Equations Example C. Solve 2 x – 2 = 4 x + 1 + 1
• 26. General Rational Equations Example C. Solve 2 x – 2 = 4 x + 1 Multiply both sides by the LCD : (x – 2)(x + 1) + 1
• 27. General Rational Equations Example C. Solve 2 x – 2 = 4 x + 1 Multiply both sides by the LCD : (x – 2)(x + 1) 2 x – 2 = 4 x + 1 ( ) * (x – 2)(x + 1) + 1 + 1
• 28. General Rational Equations Example C. Solve 2 x – 2 = 4 x + 1 Multiply both sides by the LCD : (x – 2)(x + 1) 2 x – 2 = 4 x + 1 ( ) * (x – 2)(x + 1) + 1 + 1 (x + 1)
• 29. General Rational Equations Example C. Solve 2 x – 2 = 4 x + 1 Multiply both sides by the LCD : (x – 2)(x + 1) 2 x – 2 = 4 x + 1 ( ) * (x – 2)(x + 1) + 1 + 1 (x – 2)(x + 1)
• 30. General Rational Equations Example C. Solve 2 x – 2 = 4 x + 1 Multiply both sides by the LCD : (x – 2)(x + 1) 2 x – 2 = 4 x + 1 ( ) * (x – 2)(x + 1) + 1 + 1 (x – 2)(x + 1) (x – 2)(x + 1)
• 31. 2(x + 1) = 4(x – 2) + (x – 2)(x + 1) General Rational Equations Example C. Solve 2 x – 2 = 4 x + 1 Multiply both sides by the LCD : (x – 2)(x + 1) 2 x – 2 = 4 x + 1 ( ) * (x – 2)(x + 1) + 1 + 1 (x – 2)(x + 1) (x – 2)(x + 1)
• 32. 2(x + 1) = 4(x – 2) + (x – 2)(x + 1) General Rational Equations Example C. Solve 2 x – 2 = 4 x + 1 Multiply both sides by the LCD : (x – 2)(x + 1) 2 x – 2 = 4 x + 1 ( ) * (x – 2)(x + 1) + 1 + 1 (x – 2)(x + 1) (x – 2)(x + 1) 2x + 2 = 4x – 8 + x2 – x – 2
• 33. 2(x + 1) = 4(x – 2) + (x – 2)(x + 1) General Rational Equations Example C. Solve 2 x – 2 = 4 x + 1 Multiply both sides by the LCD : (x – 2)(x + 1) 2 x – 2 = 4 x + 1 ( ) * (x – 2)(x + 1) + 1 + 1 (x – 2)(x + 1) (x – 2)(x + 1) 2x + 2 = 4x – 8 + x2 – x – 2 2x + 2 = x2 + 3x – 10
• 34. 2(x + 1) = 4(x – 2) + (x – 2)(x + 1) General Rational Equations Example C. Solve 2 x – 2 = 4 x + 1 Multiply both sides by the LCD : (x – 2)(x + 1) 2 x – 2 = 4 x + 1 ( ) * (x – 2)(x + 1) + 1 + 1 (x – 2)(x + 1) (x – 2)(x + 1) 2x + 2 = 4x – 8 + x2 – x – 2 2x + 2 = x2 + 3x – 10 0 = x2 + x – 12
• 35. 2(x + 1) = 4(x – 2) + (x – 2)(x + 1) General Rational Equations Example C. Solve 2 x – 2 = 4 x + 1 Multiply both sides by the LCD : (x – 2)(x + 1) 2 x – 2 = 4 x + 1 ( ) * (x – 2)(x + 1) + 1 + 1 (x – 2)(x + 1) (x – 2)(x + 1) 2x + 2 = 4x – 8 + x2 – x – 2 2x + 2 = x2 + 3x – 10 0 = x2 + x – 12 0 = (x + 4)(x – 3) hence x = –4, 3
• 36. 2(x + 1) = 4(x – 2) + (x – 2)(x + 1) General Rational Equations Example C. Solve 2 x – 2 = 4 x + 1 Multiply both sides by the LCD : (x – 2)(x + 1) 2 x – 2 = 4 x + 1 ( ) * (x – 2)(x + 1) + 1 + 1 (x – 2)(x + 1) (x – 2)(x + 1) 2x + 2 = 4x – 8 + x2 – x – 2 2x + 2 = x2 + 3x – 10 0 = x2 + x – 12 0 = (x + 4)(x – 3) hence x = –4, 3 However, this method of clearing the denominator might produce a solution(s) that does not work for the original fractional equation.
• 37. 2(x + 1) = 4(x – 2) + (x – 2)(x + 1) General Rational Equations Example C. Solve 2 x – 2 = 4 x + 1 Multiply both sides by the LCD : (x – 2)(x + 1) 2 x – 2 = 4 x + 1 ( ) * (x – 2)(x + 1) + 1 + 1 (x – 2)(x + 1) (x – 2)(x + 1) 2x + 2 = 4x – 8 + x2 – x – 2 2x + 2 = x2 + 3x – 10 0 = x2 + x – 12 0 = (x + 4)(x – 3) hence x = –4, 3 However, this method of clearing the denominator might produce a solution(s) that does not work for the original fractional equation. Specifically, we have to check that the answers obtained will not turn the denominator into 0 in the original problem.
• 38. 2(x + 1) = 4(x – 2) + (x – 2)(x + 1) General Rational Equations Example C. Solve 2 x – 2 = 4 x + 1 Multiply both sides by the LCD : (x – 2)(x + 1) 2 x – 2 = 4 x + 1 ( ) * (x – 2)(x + 1) + 1 + 1 (x – 2)(x + 1) (x – 2)(x + 1) 2x + 2 = 4x – 8 + x2 – x – 2 2x + 2 = x2 + 3x – 10 0 = x2 + x – 12 0 = (x + 4)(x – 3) hence x = –4, 3 However, this method of clearing the denominator might produce a solution(s) that does not work for the original fractional equation. Specifically, we have to check that the answers obtained will not turn the denominator into 0 in the original problem. In this example, both x = –4, 3 are good answers because they don’t turn the denominators to 0.
• 39. General Rational Equations We may use the cross multiplication to combine the two terms on the same side first to arrange the problem as a proportional problem.
• 40. General Rational Equations Example D. Solve 2 x – 2 = 4 x + 1 + 1 We may use the cross multiplication to combine the two terms on the same side first to arrange the problem as a proportional problem.
• 41. General Rational Equations Example D. Solve 2 x – 2 = 4 x + 1 + 1 We may use the cross multiplication to combine the two terms on the same side first to arrange the problem as a proportional problem. 2 x – 2 = 4 x + 1 + Treat the 1 as . 1 1 1 1
• 42. General Rational Equations Example D. Solve 2 x – 2 = 4 x + 1 + 1 We may use the cross multiplication to combine the two terms on the same side first to arrange the problem as a proportional problem. 2 x – 2 = 4 x + 1 + Treat the 1 as . 1 1 1 1 2 x – 2 = 4 + (x + 1) x + 1
• 43. General Rational Equations Example D. Solve 2 x – 2 = 4 x + 1 + 1 We may use the cross multiplication to combine the two terms on the same side first to arrange the problem as a proportional problem. 2 x – 2 = 4 x + 1 + Treat the 1 as . 1 1 1 1 2 x – 2 = 4 + (x + 1) x + 1 2 x – 2 = x + 5 x + 1
• 44. General Rational Equations Example D. Solve 2 x – 2 = 4 x + 1 + 1 We may use the cross multiplication to combine the two terms on the same side first to arrange the problem as a proportional problem. 2 x – 2 = 4 x + 1 + Treat the 1 as . 1 1 1 1 2 x – 2 = 4 + (x + 1) x + 1 2 x – 2 = x + 5 x + 1
• 45. General Rational Equations Example D. Solve 2 x – 2 = 4 x + 1 + 1 We may use the cross multiplication to combine the two terms on the same side first to arrange the problem as a proportional problem. 2 x – 2 = 4 x + 1 + Treat the 1 as . 1 1 1 1 2 x – 2 = 4 + (x + 1) x + 1 2 x – 2 = x + 5 x + 1 2(x + 1) = (x – 2)(x +5) You finish it. . .
• 46. General Rational Equations An answer that doesn’t work for the original problem is called an extraneous solution.
• 47. General Rational Equations Example E. (Extraneous solution) Solve 3 x – 3 = x x – 3 – 2 An answer that doesn’t work for the original problem is called an extraneous solution.
• 48. General Rational Equations Example E. (Extraneous solution) Solve 3 x – 3 = x x – 3 – 2 An answer that doesn’t work for the original problem is called an extraneous solution. Multiply the LCD (x – 3) to both sides.
• 49. General Rational Equations Example E. (Extraneous solution) Solve 3 x – 3 = x x – 3 – 2 An answer that doesn’t work for the original problem is called an extraneous solution. Multiply the LCD (x – 3) to both sides. 3 x – 3 = x x – 3 – 2( ) (x – 3)
• 50. General Rational Equations Example E. (Extraneous solution) Solve 3 x – 3 = x x – 3 – 2 An answer that doesn’t work for the original problem is called an extraneous solution. Multiply the LCD (x – 3) to both sides. 3 x – 3 = x x – 3 – 2( ) (x – 3) 1
• 51. General Rational Equations Example E. (Extraneous solution) Solve 3 x – 3 = x x – 3 – 2 An answer that doesn’t work for the original problem is called an extraneous solution. Multiply the LCD (x – 3) to both sides. 3 x – 3 = x x – 3 – 2( ) (x – 3) 1 1
• 52. General Rational Equations Example E. (Extraneous solution) Solve 3 x – 3 = x x – 3 – 2 An answer that doesn’t work for the original problem is called an extraneous solution. Multiply the LCD (x – 3) to both sides. 3 x – 3 = x x – 3 – 2( ) (x – 3) 1 1 (x – 3)
• 53. General Rational Equations Example E. (Extraneous solution) Solve 3 x – 3 = x x – 3 – 2 An answer that doesn’t work for the original problem is called an extraneous solution. Multiply the LCD (x – 3) to both sides. 3 x – 3 = x x – 3 – 2( ) (x – 3) 1 1 (x – 3) 3 = x – 2(x – 3)
• 54. General Rational Equations Example E. (Extraneous solution) Solve 3 x – 3 = x x – 3 – 2 An answer that doesn’t work for the original problem is called an extraneous solution. Multiply the LCD (x – 3) to both sides. 3 x – 3 = x x – 3 – 2( ) (x – 3) 1 1 (x – 3) 3 = x – 2(x – 3) 3 = –x + 6
• 55. General Rational Equations Example E. (Extraneous solution) Solve 3 x – 3 = x x – 3 – 2 An answer that doesn’t work for the original problem is called an extraneous solution. Multiply the LCD (x – 3) to both sides. 3 x – 3 = x x – 3 – 2( ) (x – 3) 1 1 (x – 3) 3 = x – 2(x – 3) 3 = –x + 6 x = –3 + 6
• 56. General Rational Equations Example E. (Extraneous solution) Solve 3 x – 3 = x x – 3 – 2 An answer that doesn’t work for the original problem is called an extraneous solution. Multiply the LCD (x – 3) to both sides. 3 x – 3 = x x – 3 – 2( ) (x – 3) 1 1 (x – 3) 3 = x – 2(x – 3) 3 = –x + 6 x = –3 + 6 x = 3
• 57. General Rational Equations Example E. (Extraneous solution) Solve 3 x – 3 = x x – 3 – 2 An answer that doesn’t work for the original problem is called an extraneous solution. Multiply the LCD (x – 3) to both sides. 3 x – 3 = x x – 3 – 2( ) (x – 3) 1 1 (x – 3) 3 = x – 2(x – 3) 3 = –x + 6 x = –3 + 6 x = 3 However this is an extraneous answer because it is not usable – it turns the denominator into 0.
• 58. General Rational Equations Example E. (Extraneous solution) Solve 3 x – 3 = x x – 3 – 2 An answer that doesn’t work for the original problem is called an extraneous solution. Multiply the LCD (x – 3) to both sides. 3 x – 3 = x x – 3 – 2( ) (x – 3) 1 1 (x – 3) 3 = x – 2(x – 3) 3 = –x + 6 x = –3 + 6 x = 3 However this is an extraneous answer because it is not usable – it turns the denominator into 0. Hence there is no solution for this rational equation.
• 59. General Rational Equations Ex. A. Rewrite the following fractional equations into ones without fractions. You don’t have to solve for anything. 1 4 A 1 2 B += 2 3 C1. 5 6 A 3 4 B =– 2 3 C2. 5 4 A 6 5 B –= 2C3. 1 6 A 3B =– 2 7 C4. 1 x 1 y =+5. 1 x 1 y = 1–6. 1 xy 1 x 1 y =+7. 1 (x – y) = x– 18. 1 z Ex. B. Rewrite the following equations into linear equations without fractions. Solve for the x and check the answers. 2 x =+ 19. 3 x = 110.1 2x 1 3x– 3 4x =– 111. 6 7x = 112.5 6x 7 8x –
• 60. General Rational Equations Ex. C. Rewrite the following equations into linear equations without fractions. Solve for the x and check the answers. 2 x – 1 =+ 113. 3 2x – 2 3 x – 2 =+ 214. 1 3x – 6 5 x – 2 =– 215. 1 6 – 3x 3 x = 3 x + 2 + 2 Ex. D. Rewrite the following into 2nd degree equations. Solve for the x and check the answers. 20. 3 x – 2 – 3 x + 2 = 1 21. 2 x = x+ 116. 3 x = x – 217. –6 x = x+ 518. –3 x = x – 419. 6 x + 1 = 6 x – 1 22. 12 x – 1 – 12 x + 1 = 123.
• 61. General Rational Equations 24. 1 x + 1 x – 1 = 5 6 25. 1 x + 1 x + 2 = 3 4 26. 1 x + 1 x – 2 = 4 3 27. 1 x + 2 x + 2 = 1 28. 2 x + 1 x + 1 = 3 2 29. + 5 x + 2 = 2 2 x – 1 30. – 1 x + 1= 3 2 31.6 x + 2 – 4 x + 1= 1 1 x – 2