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Math 70: Intermediate Algebra
Study Problems for the Final Exam
1. Given f ( x)  5  2 x , write f (6  h)  f (6) in simplest form.
2. Find the equation of the line that goes through the point (1,2) and is perpendicular to the
line given by the equation 2x + 3y = 6.
3. Use the addition method to solve:
3 x  2 y  3 z  8      (1)
2 x  3 y  2 z  10      (2)
x  y  z  2        (3)
x y 4
4. Graph the solution set. 2 x  3 y  6
x0
 3a b 
 2ab 
2
5. Simplify
3 2
4 3
6. Divide by using synthetic division. ( x 3  x  2)  ( x  1)
2
x4
7. Simplify
6
x 3
x4
x 1
8. [Uniform Motion Problem] Distance = Rate ● Time.
Because of weather conditions, a bus driver reduced the usual speed along a 165-mile bus
route by 5 mph. The bus arrived only 15 minutes later than its usual arrival time. How
fast does the bus usually travel? (Hint: Use hour for time.)
Distance Rate = Time
Usual condition
Current condition
2
1 1
  a.
x b
3x 2  10 x  8
10. Simplify
3x 2  14 x  8
9. Solve for x,
3
2x
 2
x  1 x  2x  1
11. Simplify
2
5
2x
5


x  7 x  12 x  3 x  4
3 y
13. Simplify
3 y
12. Solve
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14. Simplify
9b 3  25b 3  49b 3
15. Solve x  2 x  1  9
16. Simplify
17. Solve
5  3i
4  2i
x2
 0 , show your work and present your solution set in interval form.
( x  1)( x  1)
18. Describe how to obtain the graph of y  ( x  2) 2  1 from the graph of f ( x)  x 2 .
Sketch the graph.
19. Given that g ( x)  3x  2 and h( x)  x 2  1 .
Find g[h(0)] and h[ g ( x)] .
20. Find the inverse function of the function defined by the equation f ( x)  4 x  2 .
3t
1
 1
21. Given R (t )    , evaluate (a) R  
3
 3
and (b) R (1) .
22. Solve for x, 5  x 1  15 . Give the exact solution first and then round your answer to the
nearest thousandth.
23. Solve for x, log 4 x  log 4 ( x  2)  log 4 15
24. Solve for x, log( x 2  3)  log( x  1)  log 5 , give the exact answers.
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25. An isotope of technetium is used to prepare images of internal body organs. The isotope
has a half-life of approximately 6 hours. A patient is injected with 30 mg of this isotope.
(a) What will be the amount of technetium in the patient after 3 hours?
(b) How long (in hours) will it take for the amount of technetium in the patient to
1
reach 20 mg? Use the exponential decay equation: A  A0  
2
t
k
26. An investor broker deposits $2500 into an account that earns 6% annual interest
compounded monthly.
(a) What is the value of the investment after 4 years? (Round to the nearest dollar.)
(b) How long will it take her to double her investment?(Round up to a whole
number.)
n
Use the formula: P  A1  i 
 3x 2
27. Write the logarithm in expanded form: log  3 2
y z



28. Write as a single logarithm with coefficient of 1: 2 ln x  ln y  3 ln z
29. A tour operator believes that the profit P, in dollars, from selling x tickets is given by
P( x)  40 x  0.25x 2 .
(a) Find the axis of symmetry
(b) Find the vertex.
(c) Find the x-intercepts.
(d) Find the y-intercepts.
(e) Sketch the graph of the profit function.
(f) How many tickets do you have to sell in order to get the maximum profit?
(g) What is the maximum profit?
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Solution to the study final problems
1. -2h
5.
3
1
2. y  x 
3. x=6, y=-2, z=2
2
2
 9a
8b 6
6. x 2  x  2 
b
9. x 
ab  1
13.
4
x 1
x4
10.
x4
y  3  2 3y
3 y
14. 5b b
7.
4. (skip)
x2
x2
8. 60 mph
2x 2  x  3
11.
x  1x  12
15. x=5
12. x 
16.
5
2
7 11
 i
10 10
17.  ,  1 (1, 2]
18. (a) shift horizontally 2 units to the right. (b) move vertically up 1 unit.
1
1
1
19. (a) 1, (b) 9 x 2  12 x  5 20. f 1 ( x)  x 
21. (a) 3 (b)
27
4
2
22. (a) x  1 
log15
log 5
(b) x   0.683 23. x=5 24.
25. (a) 21.21 mg (b) 3.51 hours 26. (a) $3,176
x
5  33
2
(b) 11.58 years
 x2 z3 
27. log 3 +2log x -3log y -2log z 28. ln 

 y 
29. (a) x=80 (b) (80, $1600) (c) (0,0) (160,0) (d) (0,0) (f) 80 tickets
(g) $1600.