Econ 1078: Math Tool I Basic Algebraic Operations

Midterm #1 Sep. 28, 2006

Your Name: ______Your Number______

Directions:
There are 20 questions and 100 points total. Each question is worth 5 points. Give the best answer for the following questions. Show all your work instead of just putting down the answer. You can receive partial credits for the correct steps.
No graphing calculators are allowed. Please find a blank page for scratch work attached. If you need more scratch paper, please raise your hand during the exam period.
Please make your final answer clear by writing legibly and placing a box around it.
You have 75 minutes to complete the exam. Good Luck.

1) Answer True or False by writing the whole word as ‘T’ and ‘F’ can look alike (you don’t have to show your calculations here):

a. ___T_____

___F______

I t is false for the general case, you cannot cancel out (x-2). x=3 is a special case, which makes this a true equality, but in general such simplification would be false.

2) Evaluate the following numerical expressions

For the questions 3-5, simplify the expressions. Leave the results with positive exponents only

4. =

For questions 6-7 perform the given calculation by fully expanding the polynomial:

For questions 8-9 factor the polynomial completely:

Rationalize denominator:

10.

For the questions 11-12, find all x that satisfy the inequality:

11.

12.

Divide real number line into three intervals

Pick any point from each interval, i.e.

Plug each point back to your equation and evaluate it.

For ,

For

For , so only x = 0 satisfies the inequality, which means that all numbers from interval (-2,2) will satisfy the inequality, thus the answer is

-2<x<2

13. Solve for M

For the problems 14-15 solve the following quadratic equations

14.

15.

16. Solve the following equation. Don’t forget to specify variable restriction!

Set numerator = 0,

Either or

Variable restriction: Denominator can never be equal to 0, thus

Thus, the final answer:

17. After graduating from CU Boulder you decide to open your own restaurant. You think that there are not enough sandwich places on the Hill, and your new restaurant NoVeggiesSandwich will have a huge success. It will cost you $3,000 per month to rent a place, and you estimated that it cost you $2 to make each sandwich. The price of each sandwich is $4. How many sandwiches will you have to sell per month to break even?

Let X be number of sandwiches one should produce to break even.

Total Cost =

Total Revenue = 4X

Total cost = Total Revenue

= 4X

X = 1,500

Thus, for the company to break even you should sell 1,500 sandwiches.

18. You decide that during the summer you might use additional grill in your restaurant, and there are two choices for you. You can rent a grill for the cost of $40/day. Or you can buy one at $2,000 with maintenance costs around $2/day. Let X denote the number of days that the grill is needed. Find out the value of X that makes these two choices equivalent.

Let X be number of days that will make two choices equivalent.

Option 1: Cost = 40X

Option 2: Cost = $2000 + 2X

40X = 2000 +2X

X = 2000/38 days

19. Your parents decide to give you a car as a gift for getting an A in MathTools for Economists class. And you know you would sell that car in 5 years from now and keep the money to yourself. Your parents give you a choice. They could buy a car for 20,000 that depreciates each year by 12%, or they could buy a car for 18,000 that depreciates each year by 10%. If you are thinking like economics, and don’t really care about the attributes of the car, which car would you prefer to have? (Hint: you have to find the value of both cars in 5 years and make a comparison)

Value of car 1 in 5 years: V1 = 10,554.6

Value of car 2 in 5 years: V2 = 10,628.8

So comparing two values it is clear that after 5 years 2nd car would be worth more than 1st car, thus you should tell your parents to buy you second car.

20. Solve the system of equations:

In your own words explain, what does solution to the system of equation means?

If (x*, y*) is a solution to the system of two equations, it means that (x*, y*) should satisfy both equations of the system simultaneously.