Exercise 6.2. Q. 1. x = 3. Q. 2. y = 2. Q. 3. 2x = 8 x = 4. Q. 4. 3a = 27 a = 9. Q. 5. 3x = 3 x = 1. Q. 6. 4x = 20 x = 5. Q. 7.

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1 Chapter Exercise. Q.. (i) (vi) 9 (ii) 7 (vii) 9 (iii) (viii) (iv) (ix) (v) (x) Q.. (i) () + ( ) = = = False (, ) is not solution (ii) + = = () + () = + = = (,) is solution (iii) ( ) () = = = False (,) is not solution (iv) + () = + = = () () = 8 = 8 8 = 8 (,) is solution (v) + () = + = = False (,) is not solution Q.. x =, = Test: = = () + () = + = = x =, = is solution It s more complicated to use trial and error when two variables (x and ) are involved then when just one variable is involved. Exercise. Q.. x = Q.. = Q.. x = 8 x = Q.. a = 7 a = 9 Q.. x = x = Q.. x = x = Q. 7. 7x = x = Q. 8. = = Q. 9. x = x = Q.. x = 7 Q.. x = 8 x = Q.. t = t = Active Maths Book Strands Ch Solutions

2 Q.. x = x = Q.. x = 8 x = Q.. x = x = Q.. x = x x = x = Q. 7. x + = x + x = Q. 8. = 8 = = 8 Q. 9. x + = x + 7 x = Q.. x + x + = 9 x = Q.. x + = x + x x + = x x = x = Q.. x 8 = x + x = x =. Q.. x 8 + = x + x = x = Q.. x + = x + 8 8x = x = 9 Q.. a + 77 = a + a = a = Q.. x = x + x x = x = Q. 7. x = 7x + x = 8 x = 7 Q. 8. x 8 + x + = x x = x = Q. 9. = 7x x x = x + = x = x x = Q.. = + = = ( ) = + ( ) ( ) = = = is solution Q.. a = a a a = a 8a = 8 a = ( ) = ( ) ( ) = ( ) ( ) = + = a = is solution Q.. b + = b b = 7 b = 7 ( 7 + ) = ( 7 + ) ( ) = ( ) = 8 = 8 b = 7 is solution Active Maths Book Strands Ch Solutions

3 Exercise. Q.. x + = x = Q.. x = x = x = Q.. x + = x = Q.. a + = a = Q.. (x + ) = x x + = x = x x = Q.. 9x = 8 9x = x = 9 x = Q. 7. x x = x = x = Q. 8. x 8x = x = x = Q. 9. (x + ) = (x ) x + = x 7 = x x = 7 Q.. x = x = x x = Q.. (x + ) (x ) = () x + x + = x + 7 = x = Q.. ( ) + 8( ) = () + = = Q.. x (x ) = x x + = x + = x = = = Q.. (x + ) 7(x ) = ( ) x + 8x + 7 = x + = x = 7 x = 8 Q.. (x ) = 7(x + ) + () x = 7x + + x = x = Q.. (t ) + t + = () t + t + = t = t = Q. 7. (x + ) + (x + ) = () x x + = 7x + = 7x = 7 x = Q. 8. (x ) + (x ) = () x 8 + x = 9x = 9x = x = Q. 9. (x + ) + (x 9) = 7 x + + x 7 = 7 x = 7 x = x = Active Maths Book Strands Ch Solutions

4 Q.. 7(x + ) (x ) = 8x + x + = 8x + = 8x = 8 x = Q.. ( x + ) + x + = () x + + x + = x + = x = x = Q.. x + + = x + (x ) x + = x = x 9 = x Q.. (x + ) + (x ) = x + 9 x + + x 8 = x + 9 x = x + 9 x = x = Q.. x + x = x = x = Q.. (x ) (x ) = x x + = x = x = Q.. x (x + ) = 7(x ) x x = 7x 7 78 = 8x x = 9 Exercise. Q.. x = 8 x = + = 7 = x =, = Q.. x = x = = = x = 8, = Q.. x = x = = = x = 7, = Q.. x + = (x + = 9) x = + = = x =, = Q.. x + = (x + = ) x = 9 x = () + = = x =, = Q.. x = x = () + = 9 = x =, = Q. 7. x = = = x = 7, = Q. 8. x + = 9 (x + 9 = ) = = x + () = 8 x = x =, = Active Maths Book Strands Ch Solutions

5 Q. 9. x + = 8x = 9x = 8 x = () = 8 = = x =, = Q.. x + = (x + = 9) 7 = = x + ( ) = x = x = x =, = Q.. x = x = ( ) + = 7 = = x =, = Q.. x + = 8 (x + = ) = = x + 7() = 7 x = 8 x = x =, = Q.. x + = (x + 8 = ) = = x + 9( ) = x 7 = x = x = x =, = Q.. x = (x + 8 = ) = = x + () = x = x =, = Q.. x + 8 = 8 (x = 8) 7 = = x + = x = 7 x = 7, = Q.. x + = 9 (x = 8) 9 = 9 = x 7( ) = 9 x + 7 = 9 x = x = x =, = Q. 7. x = x = x + 7 = (x = ) x = x = = Q. 8. x = x + = 9x = 8 x = + = = = x =, = Active Maths Book Strands Ch Solutions

6 Q. 9. x + = (x + = ) x = () + = = = x =, = Q.. x + = (x + = ) = = x + () = x = 8 x = x =, = Q.. x + = x + = x + = (x + = ) = = x + ( ) = x = x = x =, = Q.. 7 = = x () = 8 x = x = x =, = Q.. x 8 = (x + = ) = 7 x 9 ( ) = x = x = x =, = Q.. = = x + ( ) = x + = x = x = x =, = Q.. x = x + = 7x = x + = 7 9x = 7 x = ( ) + = = = x =, = Q.. x + 9 = 8x = 9 x + = x + = 8x = 9 x = x =.. + = =. =. x =., =. = Active Maths Book Strands Ch Solutions

7 Q. 7. x + = 8 x = 8 x + = (x = 8) = 8 = x + () = 8 x = x =, = Q. 8. x = x + = x = 9x + = 9x = 7 x = () + = = = x =, = Q. 9. x + = x 7 = x + = 78 (x 7 = ) = 78 = x + () = x + = x = x =, = Q.. (x + ) + ( x) = x + = 8 x + + x = x + 7 = x + = = = 8 x + (8) = 8 x + = 8 x = x =, = 8 Exercise. Q.. (i) x = (.7, ).7.. (ii) x = (,) 7 x (iii) x = (,) 7 x (iv) x = 7 x (, ) 7 x Active Maths Book Strands Ch Solutions 7

8 (v) x = (,) x (vi) x = (,) 7 x (vii) x = x (, ) (viii) x = 8 (8,) x (ix) x = 7 ( 7,) x (x) x = x (, ) Q.. (i) x = 7, = x (.8,.7) (ii) x =, = (,) 7 8 x 8 8 Active Maths Book Strands Ch Solutions

9 (iii) x =, = (,) 7 8 x Q.. (a) (i) x =, = (,) 7 x 7 (iv) x =, = (.,.7) x (ii) x =, = (,.) 7 x (v) x =, = 7 x (, ) (vi) x =, = 7 x (, ) (b) (i) () () = = = () () = = = (,) is solution (ii) + ( ) = + = = () ( ) = (, ) = = is solution Active Maths Book Strands Ch Solutions 9

10 Exercise. Q.. 7x 7 x Q.. (i) 8, 9, 7 (ii), 7, 8 Q.. (iii) 8, 7, x> (iv),, (v),, Q.. (i) (ii) x > 7 7 (iii) (iv) Q. 7. x > x > 7 (v) (vi) (vii) (viii) (ix) x x 7 Q. 8. Q. 9. x 8 Q.. Q x N Q.. Q x x N Q.. Q x< 9 x > 8x + x > x>7 x 8 (x ) > 8x + x 7 x 7 x < x 7 7 x x + < 9 x + < x<9 x x> Q.. x Q.. 7 x Exercise.7 Q.. Q.. Q Q.. x > 7 8 (x) Active Maths Book Strands Ch Solutions 7 8 9

11 Q.. x + x x x x Q.. x + > x 9 x > x > Q.. 9(x + ) 8(x + ) 8x 7 8x x x x Q. 7. x x + x x Q. 8. x + + x + 9 x + 7 x 7 x Q. 9. (i) (ii) (iii) (iv) (v) 7 x < > x x < Q.. (i) < x < 7 (ii) x < (iii) x x (iv) 9 x < 9 9 x < x > < x (v) x + x + 8 x < x > < x Q.. x x x {,,,,, } Q.. (i) x x 7, x Z (ii) x x x, x Z (iii) E F = {,,,,,,,, 7} Q.. (i) x < 8 x <, x R (ii) x < x > x > 7, x R 8 7 (iii) 7 8 Active Maths Book Strands Ch Solutions

12 Q.. (i) x x, x R (ii) x < 9 x <, x R (iii) Q.. (i) x > x > x >, x R 7 8 (ii) x x 8 x M N = {} Q.. (i) (7x ) + (x ) > 9x 7 > 9x > 7 x > 7 (ii) x + < 8 x < 7 x < 7 (iii) Exercise.8 Q.. (i) x = (ii) = (iii) a = a = (iv) b = b = 7 (v) r = r = (vi) x = 9 x = 9 (vii) = x = x (viii) b = 7 a b = 7 a (ix) b = c b = c (x) r = t r = t (xi) x = 9 x = 9 (xii) x = x = x x (xiii) ab = 7 a b = 7 a a (xiv) b = bc b = c b (xv) r = t r = t Q.. (i) x = (ii) = (iii) a = a = (iv) b = 9b = 9b = b = 9 (v) r = r = r = r (vi) x + = x = (vii) + x = = x (viii) a b = a = b a = b Active Maths Book Strands Ch Solutions

13 Q.. Q.. (ix) 8b c = 8b = c 8b = c (x) r = r t t = r t = r (xi) + x = x = x x = x( ) = x (xii) + x = x + = x x = x x = (x ) x x = (xiii) b a = ab b ab = a b( a) = a a b = a (xiv) c b = bc c bc = b c( ) = b b c = b (xv) pr = tr pr + tr = r(p + t) = r = p + t t = n A l = w P Q.. m = v Q.. v at = u Q. 7. v u = a t k Q. 8. v = m Q. 9. ± E m = c Q.. Fr v = m Q.. r = mv F Q.. q = rs p Q.. b a = c Q.. a b = c Q.. s a c = b Q.. b(a + c) = c c b = a + c Q. 7. s at = ut s t at = u OR = s at t Q. 8. s ut = at OR a = s ut = at s t u = a OR a = t Q. 9. x + = z = z x Q.. Q.. a + b = c bc + ac = ab ac = ab bc ac = b(a c) ac a c = b A x + = z Q.. A = (x + ) z A z = x + A z = x Q.. ax bx = c x(a b) = c c x = a b Q.. pq qr = c q(p r) = c c q = p r Q.. b bx = x a b( x) = x a b = x a x s ut t s ut t Active Maths Book Strands Ch Solutions

14 Q.. (a t) = ac a ac = t a( c) = t t a = c T Q. 7. p = l g T p = g l T g p = l bc Q. 8. a = b + c ab + ac = bc ac = bc ab ac = b(c a) ac c a = b Q. 9. v = gh v h = g Q.. (i) V = M D (ii) = cm. Q.. (i) x k(a + b) = (ii) x = k(a + b) x = k(a + b) + x = k(a + b) + Q.. (i) V =. () () =,7 cm V (ii) pr = h 78 h = = cm. () V (iii) ph = r V = r (since r > ) ph 77 r = 7 77 = 7 =. =. cm Q.. (i) T = p 9.8 = (.) 9.8 =.8 seconds l (ii) T = p g T p = l g T p = g l T g p = l Exercise.9 Q.. (i) (7) = (ii) 7(7) = 9 (iii) 7n () (9.8) =. metres (.) Q.. (i) (ii) () = (iii) t Q.. (i) () + = 9 (ii) x + Q.. (i) + = (ii) x + Q.. (i) l (ii) m Q.. x + n Q. 7. (i),, 7, 8, 9 (ii) n +, n +, n +, n +, n + Q. 8. (i),,, 8 (ii) p +, p +, p +, p + 8 Q. 9. (i) = 7 (ii) x Q.. (i) = (ii) 7 = (iii) ( x) Q.. (i) 9 7 = (ii) 9 x Active Maths Book Strands Ch Solutions

15 Q.. (x) + (x ) = x + 8x = (8x ) m Q.. ( x ) for the three oungest x for the eldest Q.. (i) x (ii) (x + ) Q.. (i),, 7 (ii) q +, q +, q + Q.. cm x cm (i) Surface area = x ( ) = x (ii) Volume l = x = x b = = h = Vol = (x )( )() Exercise. Q.. x + 7 = x = 8 x = 9 Q.. x = 79 x = 8 x = Q.. x 7 = x = x = Q.. x + 7 = x = 88 x = Q.. x = x + x = x = 7 Q.. n + n + = 8 n = 8 n =, Q. 7. 8n = 7(n + ) 8n = 7n + n =, 7 Q. 8. n = n + + n + n = 7 n = 9 9,, Q = 8 = = ears old Q.. x + ( x) = 7 x = 7 x = 7 7 luxur seats Q.. x = (x + ) x x = x =, Q.. 7x = (x + ) + x = x =, Q.. x + x + + (x + ) = 9 x + 9 = 9 x = 8 x = Emil: Frances: 7 Father: () = Active Maths Book Strands Ch Solutions

16 Q.. (x + ) = (x ) x + 8 = x = x 7 = x First Rectangle: 9, Second Rectangle:, Q.. (a) 8x = x =,, 7 (b) x + x + x + = 8 x = 8 x = x =,, 8 Q.. x + x + + x + + x + = x + = x = 8 x =,,, Q. 7. x + x = (x ) x = 8x = x x = Small: kg Medium: = kg Large: () = kg Q. 8. Yellow: } + = } Yellow: Blue: + = Blue: Green: = Green: 7 Q. 9. x + x + + x + + x + + x + 8 = x + = x = x = Answer: Q.. x + x + (x) = } x + x Am: = 8x + x = Brendan: x = Chloe: x = Q.. x + x + x = 9x = 9x = x = 7 Answer: 7 Exercise. Q.. x + = (x + = ) x = x = = Q.. x + = x = x = x = = = Q.. x + = x = x = x = 8 = 8 = 7 Q.. x + = 9 x + = x + = (x + = ) x = 8 x = + = = = Soft Drink: Burger:. Active Maths Book Strands Ch Solutions

17 Q.. x + = x + =. (x + = ) x =. x =.7 Pen =.7 =. Pencil =. Q.. x + =.9 x + = 7. (x + =.8) x =. x =. Bar =. =.7 Drink =.7 Q. 7. x + = 7 x + = 7 x = + (x = ) = = x = () + x = Q. 8. x = x + = 7 x = 8 x = Alan + = 7 = 7 = Carol Q. 9. x + = 99 x + = 99 x + = (x + = 98) = 99 =. x = 99 x = Game: CD:. Q.. x + = x + 7 = (7x + 7 = 7) x = x = leather = fabric Q.. x + = x + = x + = x + = 9 (x + = ) 7x = x = = Coat: Bag: Q.. 9x + =. x + =. x + =. x + =. x + =. (x + =.9) =. =.8 x + (.8) =. x +. =. x =. Toffee:. Ice-pop:.8 Active Maths Book Strands Ch Solutions 7

18 Q.. Q.. Time = Distance Speed x + = x + = x + = 8 x + = 9 x + = 9 (x + = 9) 7x = x = + = 8 = First: m Second: m x = x = x = x = x = x = (x = ) = x () = x = x = x =, = Q.. x + = = (x ) = x x = x + = x = 7 + = x = = Karl: Eddie: = x Revision Exercises Q.. (a) (i) x = x = (ii) 7x = x = (iii) 8x = x = (iv) x = 88 x = 8 (v) = x 8 = x (b) (i) (x + ) = (x + ) x + 8 = x + = x (ii) (x + ) = (x) x + = x 8 = x 9 = x (iii) (x + ) = (x + ) + (8) x + = x x = 9 x = 7 (c) (i) x + = (x = ) 7 = = x = x = x = x =, = (ii) x = x = = (iii) x + = (x = ) 7 = 9 = 7 x 7 = x = 8 x = x =, = 7 8 Active Maths Book Strands Ch Solutions

19 (iv) x = x = = = Q.. (a) (i) x = x = (ii) x = x = (iii) = x 9 = x (iv) = x = x (v) x = 8 x = (b) (i) x <, x N x {,, } (ii) x, x N x {,,,, } (iii) x x, x N x {,, } (c) (i) n (ii) x + x + + (x + ) = 9 x + = 9 x = 7 x = Marie: ears old Q.. (a) (i) x = x = (ii) x + = x = x = x (iii) 8x + = x + x = x = (iv) x + + = x = 7x = x (b) (v) x + 8 = x 9x = 8 x = (i) + t p = q (ii) p v = t (iii) ax = b c a = b c x (iv) z = 8x z + = x 8 (v) t (n )a = a t a (vi) n = d A (vii) ph = r (c) (i) Adults = x Female adults: x (ii) x + = x + = (x + = ) = x = Q.. (a) (i) 8x 7 = x + 8x 7 = x 7 = 7x = x (ii) x + x + = x + 9 = = x = x (iii) x = x + x = x = (iv) x = x + x x = x + x = 9 x = 9 Active Maths Book Strands Ch Solutions 9

20 (v) x x = x x = x x = x = (b) (i) x + = (x + = 9) = = x + ( ) = x = x = x =, = (ii) x = x = + = = (iii) x = (x = 7) = = x ( ) = x + = x = x =, = (iv) x = (x = ) x = x = = = x =, = (v) x = 7 x + = (x = ) 7 = = x ( ) = 7 x + = 7 x =, = x = } (c) (i) Males: ( 7 ) (ii) x + = ( x + = ) = 9 = 8 x = euro coins 8. coins Q.. (a) (i) x + + = x = x (ii) (x ) = (x + ) x = x + x = (iii) x x = x = (iv) (x + ) (x + 9) = (x + 9) x + x = x 9 x = x 7x = 7 x = (b) (i) x > x > 7 (ii) x x (iii) x x (iv) 7 > x > x > x x < Active Maths Book Strands Ch Solutions

21 (v) + 7 x x x x (c) (i) x + = x = 8x = 9 8x 9 = 9x + 9 = 7x = x = 8 = 8, (ii) 7x = x + = 7x = x = x + = 7 x = 7 x = () + = } = x =, = = Q.. (a) (i) x < 8 x < (ii) x x x 7 8 (iii) x x x (iv) 8x > x + 7x > 8 x > (b) (v) x + 9 x x (i) s t = q s q = t s q = t (ii) v = pr h v pr = h (iii) v u = a s (iv) v as = u ± v as = u (v) v u = s a (vi) v( a) u = b c c = b v( a) u (c) (i) x + = x + = 9 (x + = ) 7x = 7 x = () + = = Apple: cents Orange: cents (ii) x, x +, x + } x + (x + ) + (x + ) = x + (x + ) + (x + ) = () x + x + + x + = 8 9x + = 8 9x =,, x = 7 Active Maths Book Strands Ch Solutions

22 Q. 7. (a) (i) x = (b) (i) x =, = (ii) x = (,9) (,) (iii) x = (,) x x x (, ) 7 (ii) x =, = (,) 7 (i) ( ) ( ) = 8 + = = + ( ) = 8 = = x =, = is solution (ii) ( ) + 7() = + = = x =, = is solution. ( ) + = 8 + = = The algebra method gives a more accurate answer. Reason: No difficult in reading graph accuratel is involved. x x Active Maths Book Strands Ch Solutions

23 Q. 8. (a) (i) () x x 7, x z () x x x, x z () E F = {,,,,, } (ii) () x < x <, x R () x 7 x, x R () (iii) () 9x x, x N () 8x 8x x, x N () E F = {,, } (b) (i) (x ) (x + ) (x ) = x x x + 8 = x + = = x (ii) (x + ) (x 7) = 7 x + x + 8 = 7 x + = 7 x = x = (iii) (x ) + 7(x + ) = 9(x + 8) x + x + = 9x 7 9x = 9 x = (iv) 9(x + ) = 8(x ) 8 7x + 9 = x 8 8 7x + 9 = x = x = x (c) (i) x + = x + = (x + = ) = = x = 9 cent coins: 9 cent coins: (ii) Graham: x Hilda: x Now x + = (x + ) x + = x + 8 x = Graham: Hilda: Q. 9. (a) (i) x = x + = 7 x 9 = (x + = 8) 9 = 8 = x ( ) = x = x = x =, = (ii) x = x + 8 = x 9 = (x + = ) = = x = } Active Maths Book Strands Ch Solutions

24 (b) (iii) x + + = x + = x + = 9 x + = 9 x + 9 = 7 (x + = 8) = = x + () = 9 x = x = (iv) x = 7 (x + ) 8( + ) + 7 = x 8 = x = (x 9 = ) = 99 = x ( ) = 7 x 8 = 7 x = x = 7 (v) = x + + x = x + = (x = 7) = = x () = x = (i) ax + kx = c x(a + k) = c c x = a + k (ii) r + rs = + c r( + s) = + c r = + c + s (iii) a b = ak a ak = b a( k) = b b a = k (iv) a b = c(b + ) a b = bc + c a c = bc + b a c = b(c + ) a c c + = b (v) a c = cd + d a d = cd + c a d = c(d + ) a d d + = c (c) (i) x + = x + = 7 (x + = ) = 7 x = (ii) (.) + x(.) =.( + x). +.x =. +.x.8 =.x = x Q.. (a) (i) () x x x () x x 9 x () (ii) () x x Active Maths Book Strands Ch Solutions

25 (b) (iii) () (x + ) (x + ) 8 x + x 8 x 8 x 9 x 9 () (i) a b = cx a = cx + b a = cx + b (ii) a = (7)() + = 8 = (c) Stace: x Now Darren: x x + = (x + ) x + = x + x = x = Stace: ears old Darren: ears old. Q.. (a) x + = x + 8 = (b) (x + 8 = ) } x = x = () + 8 = 8 = 9 (i) s( r) = a s sr = a s a = sr s a s = r = Q.. (a) (ii) r = = = (c) Let x = st digit = nd digit Eqn A: x + = Eqn B: x + = x + 8 x + = x = x = x = = 8 = 9x 9 = x = st digit = 7 nd digit = x = 7 x x = 9x x = x = (b) x + = x + = (c) (x + = ) x = x = = no. of person rooms x = 7 no. of person rooms x + = x 9 = (x 8 = ) = 7 = 7 x (7) = x = Active Maths Book Strands Ch Solutions

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