MATH 130i/130 College Algebra Name _____________________________________________ FINAL EXAM – Review

MATH 130i/130 College Algebra
FINAL EXAM – Review
Name _____________________________________________
1. Solve the equation for the variable x .
2  3x x  1 3x


7
3
7
2. Solve the equation.
1
7 5 2

 
3x 2 x 6 x
3. Solve the equation for e.
1 1 1
 
d e f
4. Two cars start out together from the same place. They travel in opposite directions, one of them traveling 7 miles
per hour faster than the other. After 3 hours, they are 621 miles apart. How fast is each car traveling?
5. A truck leaves Columbia traveling 45 miles per hour. Two hours later a car leaves Columbia going in the same
direction as the truck and traveling 60 miles per hour. How long after leaving Columbia will it take the car to
overtake the truck?
6. Simplify
4  80
.
2
7. Solve by factoring.
3x 2  2 x  5
8. Solve by using the quadratic formula.
6 x 2  11x  10
9. Solve by completing the square.
x 2  14 x  5  0
10. Solve the equation.
4m  5  9  16
11. Solve the equation.
3x  12  33x  1  2  0
12. Solve the inequality, and then write the solution in interval notation.
x 2  4 x  12
0
x5
13. Solve the equation.
x  1 5x  4

3x  2 3x  2
14. Solve the inequality. Then graph the solution, and write the solution in interval notation.
5
x
 4 or 5  2x  1  6
2
15. Solve the inequality. Then graph the solution, and write the solution in interval notation.
6  x  3x  10 and 7 x 14  3x  14
16. Find a and b if 7  x  2 , then a  5  2 x  b .
17. Solve the equation.
3x  5
2
4
18. Find the x- and y-intercepts of the given equation algebraically.
y  x 2  8x  20
19. Write the equation of the line that passes through the points 1,3 and 6,9 .
20. Find the equation of the line with an undefined slope that goes through the point 8,4 .
21. Write the equation of the line that is perpendicular to the line y  
3
x  9 and has the same y-intercept.
5
22. Write the equation of the line that is parallel to the line 4 x  3 y  12 and passes through the point  6,8 .
23. Consider the points P 7,5 and Q 3,3 . Find the distance between the two points. Then find the coordinates of
the midpoint.
24. A company publishes textbooks. It costs $3000 in fixed costs to set up the printing of the textbook, and it costs $1.80
for each book it prints. If the company prints x number of books, write a function C(x) for the total costs to the
company for printing the x books. Calculate the cost of printing 10,000 books.
25. For the function f ( x)  x 2  8x , find the following:
a)
f ( x  2) 
b)
f ( x) 
c)
f (7) 
26. Factor the following completely.
a. 25  144 p 2
b. 6 x2  18x  12
c. 25x2  10 x  1
d. 8 y 2  17 y  9
27. Find the domain of the function g ( x) 
x2
x 5
,and express it in interval notation.
28. Let f ( x)  ln x . Write the equation for g (x) if its graph is that of f (x) but has been reflected across the x-axis, left 5
units, and down 2 units.
x 3
1
and g ( x)  . Find the following:
4
x
a)  f  g (2)
29. Let f ( x) 
b)
g  f (4)
c) g  f (x)
d) Find the domain of g  f .
x7
.
2x  4
a. What is the domain of f ?
30. Let f ( x) 
b. Write an expression for the function f
c. What is the domain of f
1
?
d. What is the range of f ?
e. What is the range of f
31. Find f
1
1
?
( x) of the function f ( x) 
x3  5
.
4
1
( x) .
32. The following questions refer to the equation
x2  y 2  8x  10 y  5  0 .
a. What is the center of the circle?
b. What is the radius?
33. Find the real zeros or roots of the function g ( x)  ( x  5) 4 ( x  7) 2 (3x  2) . Next to each root, state the multiplicity of
that root. Then find the degree of the function and the maximum number of turning points.
34. Let h( x) 
x 9
x 2  16
a. Domain:
. Find the following:
b. x-intercept(s):
c. y-intercept:
d. Vertical Asymptote(s):
e. Horizontal Asymptote:
35. Find the inverse of the function g ( x)  7 2 x  10
.
36. Simplify. Write your final answer with no negative exponents.
 2 x y 
8x  x y 
2
2
3 3
3
0
37. Solve the following interest problems:
a. Suppose $14,000 is invested in an account earning 6% interest compounded semi-annually. How much money
will be in the account after 10 years?
b. Suppose $14,000 is invested in an account earning 6% interest compounded continuously. How much money
will be in the account after 10 years?
38. Write the logarithm in expanded form.
 3 4x  2 
log

2
 6 x

39. Write the logarithm in condensed form.
3 ln x 
1
ln y  ln(w  5)  2 ln z
4
40. Solve the equation for x .
log 5 x  25  3
41. Solve the equation for x .
8 x  128
42. Solve the equation for x . Round to 4 decimal places.
2 4 x 6  15
43. Use the change of base formula to evaluate.
log 7 25
44. Solve the following equation.
log 3 2 x  7  log 3 4 x  1  1
45. Evaluate the following expressions without the use of a calculator.
a.
6 log6
10
b. 10 log 24log3
c.
log 5 125
d.
4
ln 64
e3
46. A population of bacteria is growing exponentially according to the equation N (t )  5.6e 0.5t , where N (t ) is the number
of bacteria (in thousands) at time t (in days).
a. How many bacteria are there at time t  0 ?
b. How many bacteria are there one week later?
c. The population will be useful for scientific study once the number of bacteria exceeds 60,000. To the nearest
day, when will this happen?
47. Suppose that $3000 is invested in an account that pays interest compounded continuously. Find the amount of time it
would take for the money in the account to double at 4.25%.
48. The decay rate of a chemical is 7.1% per year. Find the half-life.
49. Solve the system of equations.
x  3y 1
2x  3y  7
50. Janet bought 135 pieces of candy to give away on Halloween. She bought two kinds of candy, paying $.24 apiece for
Blow Pops and $.18 for Tootsie Pops. If she spent $26.70 for the candy, how many pieces of each kind did she buy?
51. Mr. Jones invested a total of $30,000 in two ventures for a year. The annual return from one of them was 8%, and
the other paid 10.5% for the year. He received a total income of $2550 from both investments. How much did he
invest at each rate?
52. Solve the following inequality. Write your final answer in interval notation.
x2  x  6 .
 x  2
53. Solve the following rational inequality. Write your final answer in interval notation.
 x  4  x  5
54. Simplify the radicals or state that the root is not a real number. Assume all variables in the
radicand represent positive real numbers.
a.
14 x3  2 x 4
b.
18  75  7 3  8
c.
50x12
y8
d.
3
64c12 d 27
 0.