Unit 8 Rational Expressions and Equations Examples Introductory Algebra Page 1 of 18

Transcription

1 Unit 8 Rational Epressions and Equations Eamples Introductory Algebra Page of 8 When canceling quantities like a, to avoid the division by zero case we usually specify that a 0. a Remember that you can only cancel factors, not terms. Correct math canceling of factors: (y + z (y + z, and y + 0 Incorrect math canceling of terms: y + + y Factoring plays an important role in simplifying rational epressions. When adding or subtracting rational epressions you might have to do a lot of work. In general, you might need to factor any polynomials in the epressions get a common denominator for the rational epressions (the critical step! add or subtract using a c ± b c a ± b c simplify the numerator (this could even involve another factoring! simplify further by canceling any common terms in the numerator and denominator Be careful, show all your work, and make sure minus signs get distributed correctly; for eample, (+4 is equal to NOT +. Questions. Simplify. Simplify + y + 4y. 9ab a b (b + a.. Simplify Simplify Simplify Simplify ab ab b. 7. Simplify 4y + 9y 9y. 8. Simplify by + b ay a ay + ay + by + by. 9. Simplify Simplify Simplify ( Simplify 4 9 ( Simplify + y + y + 4y + y 4 + 8y + y. 4. Simplify y + 7y + 0y + 9y + 4y y + y.. Simplify Simplify Find the lowest common denominator for and Find the lowest common denominator for 9 and

2 Unit 8 Rational Epressions and Equations Eamples Introductory Algebra Page of 8 9. Simplify 8 cd + 9 d. 0. Simplify y + y +.. Simplify y + yz.. Simplify. Simplify 4. Simplify. Simplify. Simplify 7. Simplify y. y Simplify. 9. Simplify Simplify y. y+ + y+ y. For what values of is. Solve Solve Solve. Solve Solve not defined? 7. Solve 8. Solve 9. Solve The scale on a map is inch to miles. If the distance between two cities on the map is. inches, 4 how far apart are the two cities? 4. A recipe calls for cup of molasses. If the recipe 4 is for 8 people, how many cups of molasses would you need if you epanded the recipe to be for people? 4. Alfonse Elric and Winry Rockbell are traveling in Meico. They know a speed of 00 kph is approimately the same as mph. If the road they are traveling on has a speed limit of 90 kph, how many mph is the is the speed limit? 4. Jake is feet tall, and notices that he casts a shadow of 8 feet. At the same time, the new public scuplture of the part casts a shadow of feet. How tall is the sculpture? 44. It takes a person using a large lawn mower 4 hours to mow all the grass in the park. A person using the small mower takes hours to mow all the lawns. How long would it take two people using these two mowers together to mow the all the grass in the park? 4. A tropical fish company has two employees, Juan and Chet, who alternate cleaning the fish tanks each week. Juan takes hours to clean the tanks, but Chet takes 7 hours since he doesn t has as much eperience as Juan. For a special promotional sale, the boss (Hugo wants all the tanks cleaned Saturday morning just before the store opens. If Juan and Chet work together, how long will it take them to clean the tanks? 4. The carbon:hydrogen ratio in naphthalene is :4. If a sample of naphthalene has 4 carbon atoms, how many hydrogen atoms does it have?

3 Unit 8 Rational Epressions and Equations Eamples Introductory Algebra Page of 8 Solutions. + y + 4y + y 7 ( + y 7 and + y 0. 9ab a b (b + a ab a ab (b + a a(b + a and ab 0. Factor numerator using grouping method, look for two numbers whose product is 8 and sum is 7: 9, ( + ( + ( ( + Factor denominator: look for two numbers whose product is and sum is 7:, ( + ( + 4 Putting this into our rational epression (the colors here are used to track equivalent epressions: ( ( + ( + ( and Factor numerator using grouping method, look for two numbers whose product is and sum is 8:, ( ( ( ( Factor denominator using grouping method, look for two numbers whose product is 4 and sum is :, (4 (4 ( (4 Putting this into our rational epression: ( ( ( (4 4 and 0

4 Unit 8 Rational Epressions and Equations Eamples Introductory Algebra Page 4 of ( 4( ( 4 ( 4 and 0 (common factor (factor in numerator. ab ab b ( ab b(ab (ab b (ab and ab 0 b (common factor (factor in numerator 7. Numerator is a perfect square; denominator is a difference of squares. 4y + 9y (4 y 9y (4 y(4 + y (4 y(4 y (4 y(4 + y 4 y and 4 y y 8. This requires removing a greatest common factor from numerator and denominator, followed by factoring by grouping. by + b ay a ay + ay + by + by (by + b ay a y(ay + a + by + b (b[y + ] a[y + ] y(a[y + ] + b[y + ] (b[y + ] a[y + ] y(a[y + ] + b[y + ] [y + ] y([a + b] [y + ] ([b a] (b a y(a + b and y + 0 (common factor (common factor (recopy to identify factors (factor, cancel common factors

5 Unit 8 Rational Epressions and Equations Eamples Introductory Algebra Page of 8 9. Simplify Factor all polynomials: ( ( + two numbers whose product is 0 sum is :, 4 ( 4( + two numbers whose product is 4 sum is : 4, + 0 ( + ( 4 two numbers whose product is 0 sum is : 4, ( + ( + two numbers whose product is sum is 4:, ( + 0( 4 (simplify polynomial multiplication ( + 0( ( ( + ( 4 ( + ( + ( 4( + ( + and 4 0, + 0, Simplify Factor all polynomials: ( ( + 4 two numbers whose product is 0 sum is :, 4 0 ( ( + two numbers whose product is 0 sum is :, ( + ( + two numbers whose product is 0 sum is 7:, + 4 ( + ( two numbers whose product is sum is 4:, ( 0( (simplify polynomial multiplication ( 0( + 4 ( ( + 4 ( + ( + ( ( + ( + ( + 4 and + 0, 0, + 0. Simplify (. Factor all polynomials: two numbers whose product is 0 sum is 7:, ( + + ( + Factor by grouping ( + ( + ( ( + Difference of squares ( ( (simplify polynomial division ( ( (simplify polynomial multiplication ( ( ( + ( + ( ( + + and + 0, 0

6 Unit 8 Rational Epressions and Equations Eamples Introductory Algebra Page of 8. Simplify 4 9 ( 9. Factor all polynomials: two numbers whose product is sum is :, ( + + ( + Factor by grouping ( + ( + hey this was a perfect square! 4 9 ( + ( Difference of squares 9 ( common factor ( (simplify polynomial division ( 9 (4 9 (simplify polynomial multiplication ( ( 9 ( ( + ( + ( + ( and 0, + 0 ( +. Simplify + y + y + 4y + y 4 + 8y. Factor all polynomials (let the y tag along with the constants: + y + y + y + y + y + y two numbers whose product is sum is :, ( + y + y( + y Factor by grouping ( + y( + y ( + y( + y hey this was a perfect square! + 4y + y ( + y( + y ( + y( + y two numbers whose product is sum is 4:, 4 + 8y 4( + y common factor + y + y + 4y + y 4 + 8y + y + y + y + y (simplify polynomial division + 4y + y 4 + 8y ( + y + y ( + y (simplify polynomial multiplication ( + 4y + y (4 + 8y ( + y ( + y ( + y ( + y( + y4 ( + y ( + y and + y 0, + y 0 4( + y

7 Unit 8 Rational Epressions and Equations Eamples Introductory Algebra Page 7 of 8 4. Simplify y + 7y + 0y + 9y + 4y. Factor all polynomials: y + y y + 7y + y + y + y + two numbers whose product is 0 sum is 7:, y(y + + (y + Factor by grouping (y + (y + 0y + 9y + 0y + y + 4y + two numbers whose product is 0 sum is 9:, 4 y(y + + (y + Factor by grouping (y + (y + y + y y + y y two numbers whose product is sum is :, y(y + (y + Factor by grouping (y (y + 4y (y (y + difference of squares y + 7y + 0y + 9y + 4y y + y (y + 7y + (4y (simplify polynomial multiplication (0y + 9y + (y + y (y + (y + (y + (y (y + (y + (y (y + and y + 0, y + 0, y + 0, y 0. Simplify Factor all polynomials: ( + ( + two numbers whose product is sum is 8:, two numbers whose product is 0 sum is : 0, ( + + ( + Factor by grouping ( + ( two numbers whose product is 8 sum is 7: 8, ( 4 + ( 4 Factor by grouping ( + ( ( + ( + two numbers whose product is 9 sum is :, (simplify polynomial division ( ( 7 4 ( + + ( Simplify polynomial multiplication. ( + ( + ( + ( 4 ( + ( + ( + ( + 4 and + 0, + 0, + 0 +

8 Unit 8 Rational Epressions and Equations Eamples Introductory Algebra Page 8 of 8. The denominators are the same, so we can subtract immediately (8 + ( To find lowest common denominator we need to factor. (subtract rational epressions with common denominators 9 ( + ( + ( + (difference of squares The lowest common denominator is ( + (. I ve highlighted the overlap in red. 8. Factor everything first need two numbers whose product is 70 and sum is 9: 4, ( 7 + ( 7 factor by grouping ( + ( need two numbers whose product is 00 and sum is 0: 0, 0 ( + + ( + factor by grouping ( + ( + this was a perfect square 9 ( + ( ( + ( + ( + The lowest common denominator is ( + ( 7. I ve highlighted the overlap in red. 9. Nothing needs to be factored. 8 cd + 9 d 8 cd + 9 c d c 8 cd + 9c cd 8 + 9c cd 0. Nothing needs to be factored. (get LCD y + y + (y + (y (y + + (y (y + (y (y + + (y (y (y + y + + y (y (y + 4y (y (y + (add rational epressions with common denominators (get LCD (add (simplify numerator

9 Unit 8 Rational Epressions and Equations Eamples Introductory Algebra Page 9 of 8. Nothing needs to be factored. y + yz (z y(z + 4z yz + 4z + yz yz ( yz( (get LCD (add. Nothing needs to be factored. 4 4 (4 ( 4(4 ( 4 (4 ( 4 (4 ( 4 ( 4( ( 4(4 9 + ( 4(4 (get LCD (simplify numerator. We need to factor here. + ( + ( two numbers whose product is sum is :, + 4 ( 4( two numbers whose product is 4 sum is : 4, ( + ( ( 4( ( 4 ( + ( ( 4 ( + ( 4( ( + ( 4 ( + ( + ( ( 4 4 ( + ( ( 4 7 ( + ( ( 4 (get LCD (subtract (simplify 4. We need to factor here ( + ( + two numbers whose product is sum is 4:, + ( + ( two numbers whose product is sum is :,

10 Unit 8 Rational Epressions and Equations Eamples Introductory Algebra Page 0 of ( + ( + + ( + ( (factor ( + ( ( + ( + ( + ( ( + ( + ( ( + (factor ( + ( + ( ( + ( + ( + ( (add ( + ( + ( (simplify numerator: distribute + ( + ( + ( (simplify numerator: collect like terms ( + ( + ( + ( (simplify: factor numerator ( + ( and + 0. We need to factor here. + + ( + ( + two numbers whose product is sum is :, + ( + ( two numbers whose product is sum is :, ( + ( + + (factor ( + ( ( ( + ( + ( ( + ( + ( + ( ( + ( ( + ( + (subtract ( + ( + ( ( + ( + ( ( + ( + ( The numerator is prime. If we could factor it, we would. (get LCD (simplify numerator: distribute (simplify numerator: collect like terms

11 Unit 8 Rational Epressions and Equations Eamples Introductory Algebra Page of y y 8 + y + y y y ( y+ y ( y ( y + (y y (y + (y (y( (y + (y (y( y +, and 0, y ( ( + + ( ( (( + + ( ( and 0 ( ( ( ( ( ( + ( ( ( + and + 0 (common denominator (invert and multiply (multiply (cancel common factors (common denominator (add (invert and multiply (common denominator (invert and multiply (factor, cancel common factors

12 Unit 8 Rational Epressions and Equations Eamples Introductory Algebra Page of ( +4 ( + 4 ( + 4 ( + 4(4 ( + 4( ( 4 + and 0 (common denominator (invert and multiply 0. y y+ + y+ y y + y+ y+ y+ y+ y ( y+y+ y+ ( y+ y ( ( y + y y + y + (y + (y (y + (y + y and y + 0 y + (common denominator (invert and multiply. No denominator in the epression can equal zero. There are three denominators in the epression

13 Unit 8 Rational Epressions and Equations Eamples Introductory Algebra Page of 8. Lowest common denominator is. Multiply the equation by LCD. ( ( ( Check: 8 ( + ( so is a solution.. LCD is. Multiply the equation by LCD. ( ( Check: 4 ( + ( 4 so is a solution!

14 Unit 8 Rational Epressions and Equations Eamples Introductory Algebra Page 4 of 8 4. LCD is ( + ( 4. Multiply the equation by LCD. ( ( + ( + 4 ( 4 ( + 4 ( 4 ( 4 4( Check: ( 4/ + 4 ( 4/ 4 8/ + / 4 ( 4/ / / 4 / so 4/ is a solution.. LCD is ( + (. Multiply the equation by LCD. ( ( ( + + ( ( + ( ( ( + ( Check: (7/ + (7/ 7/ + 0/ / 4/ 7/ 7/ so 7/ is a solution.

15 Unit 8 Rational Epressions and Equations Eamples Introductory Algebra Page of 8. LCD is +. Multiply the equation by LCD. ( ( (7 ( + + ( ( + As soon as you try to check this in the original equation you will get a division by zero. Therefore is not a solution (it is an etraneous solution. Therefore, the original equation has no solution. 7. Factor polynomials. 4 ( ( + difference of squares Looking at the equation, we now see the LCD is ( ( +. Multiply the equation by LCD. ( ( ( 8 ( ( + ( ( + + ( ( + + ( ( + 8 ( + ( + 8. Factor polynomials Check: (0 4(0 (0 + + (0 0 so 0 is a solution. ( + ( Need two numbers whose product is sum is :,

16 Unit 8 Rational Epressions and Equations Eamples Introductory Algebra Page of 8 Looking at the equation, we now see the LCD is ( + (. Multiply the equation by LCD. ( ( ( ( + ( + + ( + ( ( + ( + ( + ( ( ( Check: 4 44 ( + ( + ( ( ( so is a solution 9. Factor polynomials. + ( ( Need two numbers whose product is sum is :, Looking at the equation, we now see the LCD is ( (. Multiply the equation by LCD. ( ( ( ( ( ( ( ( ( ( + ( ( ( ( ( As soon as you try to check this in the original equation you will get a division by zero. Therefore is not a solution (it is an etraneous solution. Therefore, the original equation has no solution. 40. Use 0.7 inches. Let be the unknown distance (in miles between the two cities. 4 map measurement in inches second map measurement in inches actual distance in miles second actual distance in miles 0.7 inch miles. inch Solve for. Note Units cancel. miles The distance between the two cities is 0 miles.

17 Unit 8 Rational Epressions and Equations Eamples Introductory Algebra Page 7 of 8 4. Let be the amount of molasses you need if the recipe is epanded to feed people. initial amount of molasses new amount of molasses initial number of people new number of people cups 4 8 people cups people ( To epand the recipe for people you will need 8 cups of molasses. 4. Let be the speed in mph corresponding to 00 kph. mph 00kph mph 90kph The speed limit on the road is.8 mph. 4. This is a similar triangle problem. Let be the height of the sculpture. Jake s height Length of Jake s Shadow ft 8ft ft ft 8 8 The height of the sculpture is 7. 7 ft. Sculpture s height Length of Sculpture s shadow One person with large mower takes 4 hrs to do mowing. In one hour they complete of the job. 4 Second person with small mower takes hrs to do the mowing. In one hour they complete of the job. Working together, the job will take hrs. In one hour they complete of the job

18 Unit 8 Rational Epressions and Equations Eamples Introductory Algebra Page 8 of 8 It will take them 9 hours to complete the job together. 9 hours 0 minutes. minutes 9 It will take them hours and minutes to complete the job together. 4. Juan takes hrs. In one hour he complete of the job. Chet takes 7 hrs. In one hour he complete of the job. 7 Working together, the job will take hrs. In one hour they complete of the job It will take them hours to complete the job together. hours 0 minutes minutes It will take them hours and minutes to complete the job together. 4. Let be the number of hydrogen atoms in the sample. carbon atoms 4 hydrogen atoms 4 carbon atoms hydrogen atoms The sample has 987 hydrogen atoms. 987