1.1 Sketch each of the sets A and B on two separate number lines.
1.2 Determine A ∩ B and give the answer in interval notation.
1.3 Determine A ∪ B and give the answer in interval notation.
QUESTION 2
Suppose the ratio of the number of boys to the number of girls in a school is 2 : 3. How many boys are there in this
school if there are 420 girls?
QUESTION 3
Suppose the price of a new jacket including VAT (at a rate of 14%), is R342.
3.1 Determine the price of the jacket, excluding VAT.
3.2 How much more would you have to pay for the jacket if VAT goes up by 1%?
QUESTION 4
A bus travels 200 km at a constant speed. A car travelling 10 km/h faster than the bus, completes the same distance
in 1 hour less than the bus.
4.1 How many hours does the bus take to travel the 200 km?
4.2 What is the speed of the car?
Answers & Comments
Verified answer
I can't do 1.1 since I have no means of sketching a picture here. :D
(1.2) A ∩ B is the intersection of A and B; that is, it consists of those numbers that belong to both A and B. Every real number greater than or equal to -2 is in A, but B doesn't have any real numbers smaller than or equal to -1, so the intersection must be an interval open on the left at -1. Further, while A extends forever to the right, B stops at 5, so the interval must be closed on the right at 5. Hence,
A ∩ B = (-1, 5].
(1.3) A ∪ B is the union of A and B; that is, it consists of those numbers that are in A or B. First, I want to make a note on language. The word "or" is technically ambiguous--it might implicitly include "or both" or it might not. In mathematics, the usage of "or" is invariably the former--it always implicitly includes the phrase "or both." It's very helpful to remember this. :D
So, the set A ∪ B contains those numbers that are in A or B (or both). A starts at -2 and goes forever to the right, and B is a subset of A. Hence A ∪ B = A = [-2, infinity).
(2) Let B be the number of boys and let G be the number of girls. Then the first sentence merely states that
B/G = 2/3.
Rearranging gives
B = 2G/3.
You are given that G = 420. Hence we have
B = 2 * 420 / 3 = 280.
So there are 280 boys.
This is all I have time for. I hope this helps you a bit even though I didn't answer all the questions.
2^(x +2) = 4*2^x so we've 4*2^x + 5 = 6/2^x, permit 2^x = y then we get 4y^2 + 5y - 6 = 0 (4y - 3)(y +2) = 0 so y = 3/4 or - 2 so as that 2^x = 3/4,when you consider that 2^x is often valuable we discard 2^x = - 2 so we've purely 2^x = 3/4 so x = (log(3/4))/ log 2.