Problem Solving

3. If xy =1 , what is the value of 2^(x+1)^2/ 2^(x-y)^2.
a. 2
b. 4
c. 8
d. 16
e. 32

I think the question is for the value of 2^{(x+y)^2}/2^{(x-y)^2} if xy = 1.
2^{(x+y)^2}/2^{(x-y)^2} = 2^{x^2+2xy+y^2}/2^{x^2-2xy+y^2} = 2^{4xy} = 2^4 = 16.

2. According to the directions on a can of frozen orange juice concentrate, 1 can of concentrate is to mixed with 3 cans of water to make orange juice. How many 12-ounce cans of the concentrate are required to prepare 200 6-ounce servings or orange juice?
a. 25
b. 34
c. 50
d. 67
e. 100

Solution:
It is given that 1 can of concentrate has capacity of 12 ounces.
Or 3 cans of water have capacity of 36 ounces.
Let n be the number of 12-ounce cans of concentrate required.
Now, 200 6-ounce cans have total capacity of 200*6 = 1200 ounces.
Or 12n + 36n = 1200
Or 48n = 1200
Or n = 25.

1. Which of the following inequalities has a solution set that, when graphed on a number line, is a single line segment of finite length?
a. X^4 >= 1
b. X^3 <= 27
c. X^2 > = 16
d. 2<=IxI <=5
e. 2<=3x+4<=6

Solution:
First consider a. alone.
Now x^4 >= 1 implies (x^2+1)(x^2-1) >= 0.
Or (x^2-1) >= 0
So, either x <= -1 or x >= 1.
Or a. does not mean a single finite line segment.
Next consider b.
x^3 <= 27 implies x < 3, which is again not a "finite" line segment.
Next, consider c.
x^2 >= 16.
This means x <= -4 or x >= 4.
This is again not a finite line segment.
Now, take up d.
2 <= lxl <= 5.
lxl >= 2 implies x >= 2 or x <= -2.
Now, lxl <= 5 implies -5 <= x <= 5.
Or, combining we have that -5 <= x <= -2 or 2 <= x <= 5.
Again, this is not a single finite line segment.
Lastly, consider e.
This means -2 <= 3x <= 2.
Or -2/3 <= x <= 2/3.
Now, this is a single finite line segment.

The correct answer is e.