The following are answers to Quiz 1.

a) By adding 5 to both sides, one gets x/2 = - 7 - 2x/3. Adding 2x/3 to both sides gives x/2 +2x/3 = - 7. LHS has a common denominator 6 and one gets (3x + 4x)/6 = 7x/6 = - 7. By multiplying both sides by 6 and dividing them by 7, one gets x = 6.

b)

If x - 5 is grater than or equal to 0, the equation becomes x - 5 = 4 . By adding 5 to both sides, one gets x = 9. If x - 5 < 0, the equation is - (x -5) = 4. By multiplying - 1 to both sides, the equation becomes x - 5 = -4. Adding - 5 to both sides gives x = 1. So the answers are 1 and 5.

Good Luck!!

Section:

1) The perimeter of a rectangle is 28 m long, and width is 2 m shorter than the length. What are the dimensions of the rectangle?

Answer

Set x to be the width. Then the length is x + 2. The total length of the perimeter is 2x + 2(x + 2) and is equal to 28. We have an equation 2x +2(x + 2) = 28. It is equivalent to 4x + 4 = 28. By adding -4 to both sides, we get 4x = 24; therefore, x = 6. So the dimensions are 6 m wide and 8 m long.

2) One year ago, Joe had a certain amount of money to invest. He made a bad investment and lost 2,000 dollars. But he also invested one -third of the money to gain 15 percent dividends and one-sixth at 12 percent interest. His total earning for the last year was 5,000 dollars on the investments. How much money did Joe have to begin with?

Answer

Set the total amount Joe had to be x. Then Joe's gain and loss of the year are : Gain : 0.15(x/3) + 0.12(x/6) and Loss : 2,000. So we have 0.15(x/3) + 0.12(x/6) - 2,000 = 5,000. By simplifying the equation, we get 0.05x + 0.02x - 2,000 = 0.07x - 2,000 = 5,000. By adding 2,000 to both sides, we get 0.07x = 7,000. Therefore, x = 100,000.

Good Luck!!

Answers for Quiz 3

Simplify

Answer

Solve

Answer

By subtracting x from and adding 5 to the both sides of 2x² - 3x = x - 5, we get 2 x² - 4x + 5 = 0. From the quadratic formula, we get

Solve

Answer

By subtracting 2x from and adding 1 to the both sides of the inequality, we get x £ 3. Good Luck!!

Answers for Quiz 4

Solve 2x² -3x + 1 ³ 0.

Answer

Using the quadratic formula or factoring, one gets 2x² -3x + 1 = 0 has two roots [1/2],  1. So these roots partition the real number line into three sections; x £ [1/2],   [1/2] < x < 1,   1 < x. Pick 0 for the first section and plug it into the original inquality; then we get 1 \gel 0. This is a right statement, so the first interval is a part of the answer. Similarly, we find the second section is not good, but the third is good. The answer is therefore x £ [1/2] or 1 £ x.

Graph a linear function y - 2 = 3 x - 1.

Answer

Find the x and y intercepts to be [(-1)/3] and 1, respectively. Then draw the line passing through these two points.

Given f(x) = 2x + 1,   g(x) = x² + x + 1, find (g°f) (2).

Answer

(g°f) (2) = g(f(2)) = g(2×2 + 1) = g(5) = 5² + 5 + 1 = 31. Good Luck!!

Answer to Quiz 5

The equation x² + y² + 2x - 3y + 5 = 0 defines a circle in the plane.

a) Complete square for x² + 2x in x.

b) Complete square for y² -3y in y.

c) Using (a) and (b), rewrite the equation of the circle in the form (x -x₀)² + (y - y₀)² = R², and find the center and radius of the circle.

d) Sketch the circle in the plane.

Answer

Sorry a typo. The equation should read x² + y² + 2x - 3y - 5 = 0. The answer is for the correct equation.

a) x² + 2x = x² + 2x + [(2/2)²] - [(2/2)²] = x² + 2x +1² -1 = (x+1)² -1.

b) y² - 3y = y² -3y + [(-3/2)²] - [(-3/2)²] = ( y² -3y +(-3/2)²) - (-3/2)² = (y - 3/2)² - 9/4.

c) x² + y² + 2x - 3y - 5 = (x+1)² -1 + (y - 3/2)² - 9/4 -5 = 0. Therefore, (x+1)² + (y - 3/2)² = 5 + 9/4 + 1 = [(20 + 9 + 4)/4] = 33/4 = ([Ö33]/2)². So, we have (x+1)² + (y - 3/2)² = ([Ö33]/2)².

d) The center of the circle is (-1, 3/2) and the radius ([Ö33]/2). You will have to sketch the graph for yourself.

Good Luck!!

Answers to Quiz 6

1. In order to graph the function y = 3|x-2| + 1, what function's graph does one have to know? How many units horizontally and vertically does the graph have to be shifted to get the graph of the original function?

Answer

First rewrite the given function in the form y - b = f(x-a). In this case, it will be y - 1 = 3|x-2|, and the function is y = f(x) = 3|x|. By reading off the rewritten expression, we see that the graph is obtained by shifting the graph of y = 3|x| right 2 units and up 1 unit.

2. Find if f(x) = x³ - x + 1 is odd, even or neither.

Answer

f(-x) = (-x)³ -(-x)+ 1 = -x³ + x +1. This is not equal to ±f(x) , so the function is neither even nor odd. Good Luck!!

Answers to Quiz 7

1) Suppose a line is given by an equation y = 2 x - 1. Find the equations of the parallel and perpendicular lines, passing through (1,1), to the given line.

Answer

If parallel, must have the same slope, so the slope is 2. Since it passes through (1,1), using slope-point formula, we have y -1 = 2(x-1) or equal to y = 2x -1.

When perpendicular, the slope is the negative of the reciprocal, so it is -1/2. Again since passing through (1,1), the equation is y -1 = (- 1/2)(x-1) or y = -1/2 x + 3/2.

2) At $ 30 per ticket, the huskies will fill the arena of 10,000 seats. The A.D. knows that every $ 10 increase in the price, 500 tickets will go unsold. Write the number of tickets sold, n, as a function of the ticket price, p.

Answer

(p - 30)/10 indicates the number of $10 increases from $30. Each such increase creates an additional 500 unsold tickets. Thus, the equation is n = 10,000 - 500 (p - 30)/10 or n = - 50 p + 11,500.

Good Luck!!

Quiz 9

Graph

Answer

Graph y = e^x and then shift 2 units to the left and one unit up.

Solve 64 = 2^{2x + 1}.

Answer

64 = 2⁶ = 2^{2x + 1} implies 6 = 2x +1. Therefore, x = 5/2.

Good Luck!!

Quiz 10

Do any two of the following problems. If you want to have extra points, you may choose one additional problem to do. Make sure you indicate which problem you chose. No extra points if not indicated.

Solve the equation

Answer

The equation implies that x² = 3x. Therefore, x² - 3x = x(x-3) = 0. From this we get x = 0, 3. But x = 0 is not a right answer, because log functions are defined only for positive numbers. Therefore the right answer is x = 3.

Find the value of log_1/5 (125) without using calculator.

Answer

Set y = log_1/5 (125). Then by definition, 5³ = 125 = 1/5^y = 1/5^y = 5^-y. Therefore, y = -3.

Graph y = log₂ (x-1) .

Answer

Sketch the graph of y = log₂ x , then shift one unit to right.

The sum of $ 1,000 is invested in an account with annual interest rate of 6 % compounded continuously. Approximately, how many years does it take for the initial investment of $ 1,000 to double its value? Use the formula

Answer

From the formula, we get 2000 = 1000 e^0.06t. Therefore, 2 = e^0.06t. By taking the log of both sides, we get \lu 2 = ln e^0.06t = 0.06t lne = 0.06t. So we have t = (ln2)/0.06. Use calculator to find this value is approximately 12. So the answer is 12 years.

Good Luck!!