Chapter 16 Exercise 16.1

Transcription

1 Chapter 6 Exercise 6. Q.. (i) x, x Z (iv) x > 6, x R (v) x, x Z (vi) x, x N (ii) x >, x N Q.. Solve x > 8, x N x > 8 x >, x N (iii) x, x R (iv) > x x <, x R both side Q.. Solve x 6, x R + + x 6 (v) x < 6, x Z x both side x, x R (vi) x <, x N (vii) x, x R x, x R (viii) x < 6 7 (ix) > x x >, x R 6 7 x > 6, x N Q. 6. Solve x + 8 x, x Z x x x + 8 x both side x 6, x Z x > x <, x R Q.. [swapping sides changes the order of the inequality sign] +x>8 x (x) < x 8 < + x 8 8 x + 8 x < Q.. Solve 8 < + x, x N x >, x Z Q. 7. Solve x + > 6 + x, x Z x + > 6 + x x+>6 x >, x Z (i) x, x N (ii) x, x Z (iii) x, x R Active Maths (Strands ): Ch 6 Solutions

2 Q. 8. Solve x x +, x R x x + x + x x + x 8 x, x R 6 Q. 9. Solve x + < x, x R x + < x x > x + x > x > 6, x R Q.. Solve x 6x, x N x 6x 6x x x x x, x N Q.. Solve x < x, x Z x < x x > x x >, x Z 6 Q.. Solve 8x > 9x, x Z 8x > 9x 9x < 8x x < x <, x Z Q.. Solve x 6 < x +, x R x 6 < x + x + > x 6 7x + > 6 7x > 7 x >, x R Q.. Solve (x ) (x + ), x R (x ) (x + ) 6x x + 6x x + x + 6x 9x + 9x 7 x, x R Q.. Solve (x ) > (x ), x R (x ) > (x ) x 8 > 6x 6x < x 8 x < 8 x < 7 x <, x R ½ Q. 6. Solve (7x ) (x + ) + x >, x N. x x 8 + x > x > x > x >, x N Q. 7. Solve (x ) 6x (x + ), x Z (x ) 6x (x + ) 6x 6x x 6x x x x x, x Z Active Maths (Strands ): Ch 6 Solutions

3 Q. 8. Solve x (6x ) < (x + ), x R x (6x ) < (x + ) x x + < x + x + < x + x + < x < x >, x R Q. 9. (i) p = number of people attending the meal budget =,7 fixed cost = cost per person = p +,7 (ii) Solving p +,7 p +,7 p, p 9 7 The maximum number of guests possible is 9. Q.. (i) 7x money withdrawn after x weeks (ii) After weeks, 7 =, = 7 7 left (iii) After x weeks, 7x left (iv), 7x < (v) Solve (iv), 7x < 7x < 6 x > 8 Siobhán will be on tour for 8 weeks Q.. Option B is cheaper, i.e. less than Option A Let x be the number of days. 7x + 76 < x + x < x < Option B is cheaper than A for the first days (note at days they are the same price and after that Option A is cheaper). Q.. (i).99x + (ii) Solve (i).99x +.99x x.6 Evan can buy books Q.. Company B is cheaper (i.e. less than company A). (i) 8x + 6 < x + 7x + 6 < 7x < x < 7 7 It is cheaper to rent from Company B for the first 7 days. Active Maths (Strands ): Ch 6 Solutions

4 (ii) Day Company A ( ) Company B ( ) (iii) Charge ( ) + = = 6 + = = Days Company A Company B (iv) 7 days as in (i) Q.. Statement Always true Never true Sometimes true If a > b and b > c, then a > c If a < and b <, then a < b If a > b, then a > b If a > b and b < c, then a < c If a + >, then a > If b < b 8, then b > If a and b are both positive and a < b, then a < b Suggested answers for Q student answers may vary If a > b, b > c then a > c. Using a numberline c b a b > c a > b a is always greater than c If a < and b < then a < b. a < a > then a > and b < a From the numberline a < b is never true. If a > b then a > b. Multiplying by a negative changes the direction of the inequality b Active Maths (Strands ): Ch 6 Solutions

5 if a > b then ( ) a < ( ) b i.e. a < b a > b is never true. If a > b and b < c then a < c. Using a number line b * * c a > b a Two possible locations for c show that a is sometimes less than c. If a + > then a >. Solving a + > a > a > a > always true c If b < b 8 then b >. b < b b < b > b > always true If a, b both positive and a < b, then a < b. Let a = and b = then also if a = and b = then i.e. true. thus a < b never Exercise 6. Q.. (i) x, x Z Q. 6. < x +, x N (ii) x, x N (iii) x, x Z (iv) x, x R < x + < x, x N (v) 8 < x <, x R (vi) < x, x R (vii) 8 x <, x R Q.. < x < 8, x N Q. 7. < x <, x Z < x < Q.. x, x Z < x < < x < < x <, x Z Q.. x <, x R x x, x R Q x, x R x 6 x Q.. x <, x R x < x <, x R x, x R Active Maths (Strands ): Ch 6 Solutions

6 Q x + < 9, x R 9 x + < 9 x < x <, x R Q.. < + x <, x R < + x < < x <, x R Q.. < 8x + 8, x N < 8x < 8x 6 8 < x, x N Q.. > x >, x R Multiplying by will reverse the direction of the inequality signs < x <, x R Q.. x, x Z x x, x Z Q.. x +, x R x + 6 x 6 x x, x R Q.. 6 7x > 6, x R x > 6 7x > 7x < 7 x <, x R Q. 6. > x 7, x N > x 7 > x < x, x N Q. 7. (i) A = {x x 6 < 9, x Z} x 6 < 9 x < x <, x Z A = {,,, } (ii) B = {a x +, x Z} x + x x, x Z B = {,,,, } (iii) A B = {,,,,, } (iv) Q. 8. (i) C = {x x 8 6, x N} x 8 6 x x 7 C = {,,,,, 6, 7} 6 Active Maths (Strands ): Ch 6 Solutions

7 (ii) D = {x x, x N} + + x x 6 x D = {,,, } (iii) C D = {,,,, 6, 7} (iv) Q. 9. (i) E = {x x + >, x R} x + > x > (ii) F = {x x + > x +, x R} 6 x + > x + (iii) E F x x x > x + x > x < Q.. (i) G = {x (x + ) > (x ), x R} (x + ) > (x ) 6x + > x + 6x > x +x 7x > 7 x > (ii) H = {x (x + ) (x ), x R} (x + ) (x ) 6 6 x + 6 x x x x x 9 (iii) G H Q.. (i) < x, x R < x + + x > x > (ii) x 9 7, x R x 9 7 x 6 x x Integer values of x which satisfy (i) x > and (ii) x are {,, } Q.. (i) t (ii) t t 8 The temperature of the water before heating could have ranged from C to 8 C. Q.. (i) Foothill zone < x, (ii) Subalpine zone,7 < x, (iii) Snow zone, < x, Active Maths (Strands ): Ch 6 Solutions 7

8 Q.. (i) 7 x dimpled ball (ii) 7 x x 6 smooth ball (iii) Range of dimples y y (iv) y < Q.. V = 8 t < 8 t < 8 < t < 8.8 < t <.8.8 > t >.8 Between.8 and.8 seconds Q. 6. (i) < p + < 9, x Z (ii) < p + < 9 7 < p <. < p <, x Z p can range from to Q. 7. (i), x + 9, (ii), x + 9,, x 9, x The least number of days is and the most is. Q. 8. (i) Smallest distance: 6 = km (ii) Greatest distance: + 6 = 6 km (iii) x 6 (v) Q. 9. (i) 8 = less sold If sold, original total is + = 6 (ii) c = 6 8n where c = number sold and n = number of weeks (iii) 6 8n (iv) 6 8n 9 8n 9 8n 6. n Therefore she was content on the rd, th, th and 6th Saturdays after the first Saturday. Q.. (a) (i) Solve < x + 8, x R (ii) < x + 8 < x < x, x R (b) < x (must spend ). < y 6 + voucher = 6 Q.. (i) 8x (ii) x 8 8x 8 x questions must be answered correctly to guarantee victory. Amy Shokri km 6 km 6 km Shokri Office 8 Active Maths (Strands ): Ch 6 Solutions

9 Revision Exercises Q.. (i) Solve x x +, x Z x x + x x + x x, x Z (ii) Solve x +, x R x + x x, x R (iii) Solve x + <, x N x + < x < 9 x <, x N (vii) Solve < x, x R < x < x > x x <, x R Q.. (i) Solve x +, x N x + x x, x N (ii) Solve < x, x Z < x < x < x 7, x Z (iv) Solve x + x 7, x R x + x 7 x 7 x x x, x R 6 7 (iii) Solve < x + 7, x R < x + 7 < x < x, x R 6 (v) Solve + x x, x N + x x x + x x + x x x, x N (vi) Solve x > 9, x R x > 9 x > x < 6, x R 6 7 (iv) Solve 9 x, x Z x 8 x x, x Z Active Maths (Strands ): Ch 6 Solutions (v) Solve < x 7 <, x N < x 7 < 8 < x < < x < 6, x N 6 9

10 Q.. (i) A: x 8, x N x 8 x 9 x A = {,, } (ii) B: + 7x <, x Z + 7x < 7x < 8 x < B = {,,,,, } (iii) C: < x + < 6, x Z < x + < 6 < x < < x < C {,,,,,, } (v) D: 8 7x, x Z x 7 7x x D = {,,,, } (vi) E: < x, x Z < x 6 < x > x OR x < E = {,,, } Q.. (i) Solve x + <, x N x + < x < 8 x < The set of positive integers less than four. (ii) x +, x Z x + x x The set of integers greater than or equal to negative one. (iii) < x + <, x R < x + < < x < < x < The set of real numbers greater than negative one but less than five. (iv) Solve x <, x R x < 6 x < x < x > i.e. < x The set of real numbers greater than a half but less than or equal to three. (v) Solve < x, x N < x < x 7 < x 8 The set of positive integers greater than ten but less than or equal to forty-eight. Q.. E: x 7, x N + + x 7 x 8 x, x N E = {,,, } F: x, x N x x x F = {,,,, } E F = {,,, } Active Maths (Strands ): Ch 6 Solutions

11 Q. 6. (i) P: 6x < 6, x R 6x < 6 6x < 6x 6 > 6 x >, x R ½ (ii) Q: x, x R x x x, x R (iii) P Q ½ 6 6 Q. 7. If x y, where x, y R (i) + + x y x y True (ii) x y x y true x (iii) y Multiplying both sides by gives x y x y Not true (iv) x y x y True Q. 8. If x > y and x, y R. (i) x + > y + x > y True (ii) + + x > y x > y True (iii) x > y x > y True (iv) x > y x < y x > y Not true If x > y, is x > y always? No, e.g. if x = and y = then x =, y = and Q. 9. Why is it incorrect to write < x <, x R? Because this means that x is less than but greater than, which is impossible. Q.. Hamza gets more than but less than. Possible amounts are:., 6., 6., 7., 7., 8., 8., 9., 9. Q.. Séamus 78 cm Nóirín 67 cm (a) Susie Nóirín Séamus 67 cm 78 cm (b) Susie s height has an infinite range of possibilities, as height is a continuous variable. Q.. (x 6) < x + x < x + x < Bill must be younger than Q.. Perimeter: (x ) + x < (x ) + x < 6 < x + x < < 6x < 6 < 6x < 8 6 < x < Area: x(x ) Active Maths (Strands ): Ch 6 Solutions

12 if x = Area = ( ) = = 6 if x = Area = ( ) = = The area is greater than 6 but less than square units Q.. (i) x + 9 > ; this is because the two CDs and the DVD cost more than the Sheena had. x + 9 > x > x >. (ii) x + 7 this is because Sheena could afford both CDs and the perfume from the. x + 7 x x 6. (iii) From (i) CD >.. From (ii) CD 6.. Q.. x + x + > x + > x > 9 As the CD price is a whole number, a CD costs 6.. x >. x = (given x is a whole number) Q. 6. (i) (ii) x, x Z x, x Z Students answers will vary. Q. 7. Width = x, length = x + Perimeter: (x + ) + x = x x = x + 8 Assuming width > < x + 8 < < x < < x <. The width is greater than but less than. cm. Q. 8. Sid s amount = x Fionn s amount = x + x + x + < 6 x + < 6 x < 8 x < Sid has less than. Active Maths (Strands ): Ch 6 Solutions