Unit 4: Rational Expressions and Equations Assignment Booklet 4 Name: < Gagandeep brar > Date: < > Unit 4: Rational Expressions and Equations Unit Assignment Read the course material and complete the practice questions in the Unit 4 Rational Expressions and Equations Guide for Learning Booklet before working on this Unit Assignment. The following chart shows you which lesson to review if you're having difficulty with the questions in this assignment booklet. Unit Assignment Questions Lesson 1, 2, 3, 4 4A 5, 6, 7 4B 8 4C 9, 10, 11, 12, 13 4D 14 *Logic and Reasoning *Contact your teacher if you need help with Logic and Reasoning. For full marks, show all calculations, steps, and/or explain your answers. Total Marks: 76 Type your answers within the blue brackets < >. They expand as you type. 1. Two students were asked to determine the non-permissible value(s) for the rational expression. 2x 5 4 x 2 10 x Student A says that the non-permissible value is 0 and Student B says that the non-permissible 5 values are 0 and . 2 Which student is correct? Explain your answer. (2 marks) ADLC Mathematics 30-2 1 Assignment Booklet 4 2. Match the rational expression in the left column with the non-permissible value(s) in the middle column. Record your answer in the right column. (4 marks - 0.5 marks each) Rational Expression Non-Permissible Value(s) Matching a. 4x2 8x i. x 2, 2 a. < vii> b. x2 x 1 ii. x 1, 0 b. < iii> x 1 1 x iii. x 1 d. 3x 2 2x 4 iv. x 0, 2 d. < v> e. 2x 3 x2 4 v. x2 e. < i> f. 2x 1 2x2 4x vi. x 1, 0,3 f. < iv> g. x 1 x 3 x 1 2x vii. x0 g. h. x3 x 3 x 1 2x viii. x 1 h. < vi> c. 2 Unit 4: Rational Expressions and Equations c. < viii> ADLC Mathematics 30-2 Unit 4: Rational Expressions and Equations Assignment Booklet 4 3. Simplify. State the non-permissible value(s). a. c. e. 4. 12a 3b2c 4 (2 marks) 16a 5b < non-permissible values, a, b = 0 4*3*a*a*a*b*b*c^4 4*4*a^2*a*a*a*b 3bc^4/ 4a^2, a, b = 0> 5b2 15b (2 marks) 10b 30 < non-permissible value, b = 3 5b(b-3)/ 10 (b-3) 5b/10, b = 3 > 14 2n (2 marks) n 2 7n b. d. f. 2 x 12 (2 marks) ( x 6)(2 x 1) < non-permissible value, x = -6, -1/2 2(x+6)/ (x+6) (2x+1) 2 / 2x+1, x = -6, -1/2 > 2m 6 (2 marks) 2m 2 18 < non-permissible values, m = 0, 9 2(m+3)/ 2(m^2-9) 2(m+3) / 2(m+3)(m-3) 2 / m-3, m = 0, 9 > 6 y2 9 y (2 marks) 12 y 2 A student made an error while trying to simplify the following rational expression, as shown below. a. Identify and explain the student's error. (1 mark) < student should find non-permissible values first before solving, even non- permissible value is correct but it seems he found permissible value from the numerator, 5x. Also, factor the expression first and then find the common values in the numerator and denominator. > b. Correctly complete the simplification. (1 mark) > 5. Consider the following table: Possibilities for a x2 4 Possibilities for b x2 4 Possibilities for c 2x 4 2x2 8 2x2 8 2x 4 2x 4 x 2 25 x 2 25 2 x 10 2 x 10 2 x 2 50 2 x 2 50 2 x 10 2 x 10 Select one expression from each column in the chart so that Show work to demonstrate the simplification. (2 marks) <> 6. Possibilities for d 2x 4 a c x2 can be simplified to . b d x 5 Find the product of each of the following. State the non-permissible value(s). a. 7 x 2 y 5 2 xy 2 (2 marks) x3 21 y < x, y = 0 7y^5/x * 2xy/ 21 b. 6ab 2 5a 3 3 (2 marks) 20b c <> 14x.y^6 / 21x c. 2y^6 /3, x, y = 0 > a 5 2a 6 (2 marks) a3 4a < non-permissible values a = 0, -3 (a+5/a+3) * 2(a+3) / 4a 2(a+5) / 4a 2(a+5)/ 2(2a) a + 5 / 2a , a = 0, -3> 4 d. x 4 10 x (2 marks) 5 x 2 80 x 1 < non-permissible values, X = 0, -1, 4, -4 (x-4) / 5(x+4) (x-4) * 10x/x+1 10 / 5(x+4) (x+1) 2*5 / 5(x+4) (x+1)) 2/(x+4) (x+1) , x = 0, -1, 4, -4 > ADLC Mathematics 30-2 Unit 4: Rational Expressions and Equations Assignment Booklet 4 7. Find the quotient of each of the following. State the non-permissible value(s). a. 6 x3 y 9x2 2 3 (2 marks) 4 xy 20 x y (2*3*x^3*y/2*2*x*y) / {(3*3*x^2/5*2*2x^2y^3} 3x^2/2 * {-(5*2*2*y^3) / (3*3)} X^2- 10y^3 / 3, x ,y = 0 > c. 8. 9 x2 x 3 (2 marks) 8x 8x <> Simplify. State the non-permissible value(s). 4 2 a. (2 marks) 2 5a 6a <> c. x2 4 (2 marks) 2 2 x 20 x 3x 30 <> d. b. d. x( x 4) 6 x 2 24 x (2 marks) x 1 2 <> 3x 2 4 x 3 (2 marks) x 1 x 1 <> 3x 3 5 (2 marks) 2 x 1 x 1 <> 9. Algebraically solve and verify each rational equation. State the non-permissible value(s). (12 marks) Equation 3 5 a. x 1 2x 1 b. 9x 2 7 2x 1 2 ADLC Mathematics 30-2 Solve (2 marks each) < By cross multiplying we get 5(x - 1) = 3 (2x - 1) So 5x - 5 = 6x - 3 So 6x - 5x = -5 + 3 So x = - 2 * Verify (1 mark each) < Verification: substitute x = - 2 in each given ratio and you will get the same answer -1 * non-permissible values: Set x - 1 = 0 so x = 1 And 2x - 1 = 0 => x = 1/2 > > < By cross multiplying : => 7(2x + 1) = 2(9x + 2) => 14x + 7 = 18x + 4 < Verification: substitute x = (3/4) in (9x+2) /(2x+1) you 5 Assignment Booklet 4 Equation x 1 5 x 20 c. x 3 x 2 16 d. x 2 1 x 1 x x 1 Unit 4: Rational Expressions and Equations => 18x - 14x = 7 - 4 So 4x = 3 => x = 3/4 > will get 3.5 = 7/2 * non-permissible values: Set 2x + 1 = 0 , => x = -1/2 > Solve (2 marks each) < By cross multiplying : (x + 3)(5x + 20) = (x-1) (x^2 - 16) So 5(x+3)(x+4) = (x-1)(x-4)(x+4) => 5(x+3) = (x-1)(x-4) So 5x + 15 = x^2 - 5x + 4 So x^2 - 10x - 11 = 0 => (x - 11)(x + 1) = 0 Hence x = 11 , x = -1 > Verify (1 mark each) < Verification: substitute x = 11 , x = - 1 in both sides you will get the same results. For x = -1 you will get -1 for both sides. And for x = 11 you get 5/7 for both sides . * non-permissible values: Set x + 3 = 0 , => x = -3 And x^2 - 16 = 0 , => (x+4)(x4) = 0 , so x = 4 , - 4> < Verification : substitute x = 2 you get value 1 for both sides * non-permissible values: Set x - 1 = 0 , so x = 1> < By taking common denominator in the lest side we get (x2 -2(x - 1) ) / [x(x-1)] = 1 / (x - 1) ... cancel (x-1) for both sides S0 [ (x2 - 2x + 2 ) / x] = 1 Cross multiplying : => x2 - 2x + 2= x => x2 - 3x + 2 = 0 => (x - 2)(x - 1) = 0 So x = 2 And x = 1 (denied , it's nonpermissible) > 10. Consider each scenario: a. A high-speed passenger train in Europe completes an 800 km trip in 3 hours. The train travels the first 600 km at an average speed that was 100 km/h faster than the last 200 km of the trip. If x represents the average speed of the train on the second part of the trip, then an equation to represent this situation is: 600 200 3 x 100 x Identify all restrictions on the variable x in this context. (2 marks) < x= 0 and x= -100 > b. 6 Kailey is making earrings to sell at a local market. She spends $80 on supplies and decides to keep 4 pairs of earrings to give to her friends. Kailey sells the rest of the earrings she made for $175, making a profit of $9 per pair of earrings. ADLC Mathematics 30-2 Unit 4: Rational Expressions and Equations Assignment Booklet 4 If n represents the number of pairs of earrings Kailey made, then an equation that represents this information is: 175 80 9 n4 n Identify all restrictions on the variable n in this context. (2 marks) < n = 4 and n= 0 > 11. Alex and Jamie work for a landscape company during the spring and summer. When Alex works alone, it takes him 5 hours to complete a spring clean-up in a yard. When Alex and Jamie work together, it takes them 3 hours to complete a yard. a. Define a variable to represent the time it takes Jamie to complete a yard alone. (1 mark) b. Write an equation to represent the time it takes Alex and Jamie to complete a yard together. (1 mark) < 1/x+ 1/5 = 1/3 > c. Solve the equation to determine the time it takes Jamie to complete a yard alone. (2 marks) < 1/x+ 1/5 = 1/3 multiply by 15x 15+.3x=5x 5x-3x=15 2x=15 x=7.5 Answer is 7.5 hours > 12. At the end of Art class, the students must clean up their work stations. When working together, Nicki and James can clean their work station in 12 minutes. When they clean the work station independently, it takes James 10 minutes longer than Nicki. Write and solve an equation to determine the length of time that it takes each of them to the work station on their own. (4 marks) < let nicki take x to clean work station alone then james take to do it alone= x+10 (1/(x+10) + (1/x) =1/12 (x+x+10)/x(x+10) =1/12 (2x+10)/x^2+10x =1/12 12(2x+10) = x^2+10x 24x +120= x^2+10x x^2+10x-24x-120 =0 x^2 -14x -120 =0 (x-20) (x+6) =0 x=20 or x = -6 ADLC Mathematics 30-2 7 Assignment Booklet 4 Unit 4: Rational Expressions and Equations x cannot be negative so, x =20 ....................answer answer is nicki take 20 hours to do alone and james take 20+ 10 = 30 hours to do on his own > 13. Amber has entered a cross country running series. Each race in the series has a time limit by which runners must complete the course in order to qualify for the next race. The first race is 9 km long and must be completed in 50 minutes. After Amber runs the first 5 km, her coach informs her she must increase her speed by 2 km/h for the last 4 km in order to finish the race in 50 minutes. a. Complete the following chart. (3 marks) Distance (km) Speed (km/h) Time (h) First 5 km Section < 5> < x> <5> Last 4 km Section <4 > < x+2> < 4/x+2> b. Write and solve a rational equation to determine the speed at which Amber needs to run the last 4 km of the race. Convert time in minutes to time in hours. (2 marks) < 5/x + 4/(x+2) = 50/60 x =10 Thus, Amber speed must be 10+2 =12 km/h > 14. Patterns and Games: Continue the Pattern The following Diagrams form a pattern. Original Diagram Diagram 1 Diagram 2 Diagram 3 Calculate the total number of new circles created in Diagram 5. (1 mark) < 40> 8 ADLC Mathematics 30-2 Unit 4: Rational Expressions and Equations Assignment Booklet 4 End of Assignment ADLC Mathematics 30-2 9