(i) y = k/(k + w); k = 1/2; w = 1/3
y = (1/2)/(1/2 + 1/3)
y = (1/2)/(3/6 + 2/6)
y = (1/2)/5/6)
y = 1/2 * 6/5
y = 6/10
y = 3/5
(ii) x/(x - 1) - 1 =
= x/(x - 1) - (x - 1)/(x - 1)
= [x - (x - 1)]/(x - 1)
= (x - x + 1)/(x - 1)
= 1/(x - 1)
(iii) 4^(2x + 1) = sqrt(8)
4 = 2^2, and 8 = 2^3
(2^2)^(2x + 1) = sqrt(2^3)
2^(4x + 2) = (2^(3))^(1/2)
2^(4x + 2) = 2^(3/2)
4x + 2 = 3/2
4x = 3/2 - 2
4x = -1/2
x = -1/8
(iv) x - 3 = sqrt(3x - 11)
Square both sides.
(x - 3)^2 = ( sqrt(3x - 11) )^2
x^2 - 6x + 9 = 3x - 11
x^2 - 9x + 20 = 0
(x - 4)(x - 5) = 0
x = 4 or x = 5
Check:
For x = 4
x - 3 = sqrt(3x - 11)
4 - 3 = sqrt(3 * 4 - 11)
1 = sqrt(12 - 11)
1 = sqrt(1)
1 = 1
1 = 1 is a true statement, so x = 4 is a valid solution.
For x = 5
x - 3 = sqrt(3x - 11)
5 - 3 = sqrt(3 * 5 - 11)
2 = sqrt(15 - 11)
2 = sqrt(4)
2 = 2
2 = 2 is a true statement, so x = 5 is a valid solution.