Math 095 - Intermediate Algebra - Prairie State College

Math 095 - Intermediate Algebra
Final Exam Review
Objective 1: Determine whether a relation is a function. Given a graphical, tabular, or
algebraic representation for a function, evaluate the function and find its domain and
range.
1. The graph of a function is shown below. Which one of the following choices represents
the domain of the function (Follow-up question: What is the range of
the function)
2
1
-2 -1 1 2
-1
(a) (−∞, −1]
(b) (−∞, ∞)
(c) (−∞, −1) ∪ (−1, ∞)
(d) (−∞, 1) ∪ (1, ∞)
-2
2. Which of the following tables describes a function
(a)
x f(x)
-1 1
0 0
1 1
(b)
x f(x)
1 -1
0 0
1 1
(c)
x f(x)
1 -1
1 0
1 1
(d) None of the tables represents a function.
1

3. Given the function f(x) = x 2 + x + 2, which of the following represents f(−1)
(a) −x 2 − x − 2
(b) x 2 − x + 2
(c) 2
(d) 0
Objective 2: Given the graph of a line, an equation of a line, or two points on a line, find
the slope and y-intercept of the line.
4. Find the slope and y-intercept of the line described by the equation 5x −2y = 10.
(a) slope is 5/2, y-intercept is (0, −5)
(b) slope is 5/2, y-intercept is (0, 5)
(c) slope is −5/2, y-intercept is (0, −5)
(d) slope is −5/2, y-intercept is (0, 5)
5. A line passes through the points (2, 0) and (4, 4). Find the slope and y-intercept
of the line.
(a) slope is 2, y-intercept is (0, −4)
(b) slope is 2, y-intercept is (2, 0)
(c) slope is 1/2, y-intercept is (0, −4)
(d) slope is −1/2, y-intercept is (−4, 0)
6. Find the slope and y-intercept of the line shown below.
y
✻
✕
✲
x
☛
(a) slope is −2, y-intercept is (0, 2)
(b) slope is 2, y-intercept is (1, 0)
(c) slope is 2, y-intercept is (0, −2)
(d) slope is 1/2, y-intercept is (0, −2)
2

Objective 3: Find equations of lines in slope-intercept or point-slope form. Find equations
of horizontal or vertical lines. Determine whether lines are parallel, perpendicular, or
neither.
7. A line passes through the points (2, −2) and (4, 1). Find an equation of the line
in slope-intercept form.
(a) y − 1 = 3 (x − 4)
2
(b) y = 2 3 x − 4
(c) y + 2 = 2 (x − 2)
3
(d) y = 3 2 x − 5
8. Find an equation of the horizontal line passing through the point (−3, 8).
(a) 8y − 3x = 0
(b) y = 8
(c) −3x = 8y
(d) x = −3
9. Two linear equations are given below. Which one of the following is true of the
graphs of the equations
y = 5x − 1
x + 5y = 10
(a) The graphs are parallel lines.
(b) The graphs are perpendicular lines.
(c) The graphs intersect at the point (0, −1).
(d) The graphs are parabolas.
Objective 4: Determine whether a system of two linear equations in two variables has no
solution, exactly one solution, or infinitely many solutions. Solve systems of two linear
equations in two variables by using substitution, elimination, or graphical methods.
10. Solve the following system of linear equations. What is the y-coordinate of your
solution
y = 2x + 4
(a) y = 4
(b) y = 1
(c) There are infinitely many solutions.
(d) There is no solution.
6x − 3y = 1
3

11. Solve the following system of linear equations. What is the x-coordinate of your
solution
6x + 3y = 12
(a) x = 6
(b) x = 4
(c) There are infinitely many solutions.
(d) There is no solution.
y = −2x + 4
12. Solve the following system of linear equations. What is the y-coordinate of your
solution
2x + y = 4
(a) y = 1
(b) y = 2
(c) There are infinitely many solutions.
(d) There is no solution.
3x + 2y = 7
Objective 5: Solve application problems that require setting up and solving a system of
two linear equations in two variables.
13. A parking meter contains only nickels and dimes worth $6.05. If there are eightynine
coins in all, what is the value of the nickels alone
(a) $57.00
(b) $3.25
(c) $5.55
(d) $2.85
14. A chemist has two concentrations of hydrochloric acid in stock: a 50% solution
and an 80% solution. How much of each should she mix to obtain 100 milliliters
of a 68% solution
(a) 40 milliliters of 50% solution and 60 milliliters of 80% solution
(b) 60 milliliters of 50% solution and 40 milliliters of 80% solution
(c) 68 milliliters of 50% solution and 32 milliliters of 80% solution
(d) 32 milliliters of 50% solution and 68 milliliters of 80% solution
4

Objective 6: Find unions and intersections of intervals. Sketch the graph of an interval.
Convert between interval notation and inequality notation.
15. Find the union and the intersection of the intervals (−∞, 6] and (−3, ∞).
(a) union: (−3, 6), intersection: (−∞, ∞)
(b) union: (−∞, ∞), intersection: (−3, 6]
(c) union: (−∞, −3), intersection: [6, ∞)
(d) union: (−∞, ∞), intersection: (−3, 6)
16. Which one of the following intervals is described by the inequality −2 ≤ x < 17
(a) (−2, 17]
(b) (−∞, −2] ∪ (17, ∞)
(c) [−2, 17)
(d) (−∞, −2] ∩ (17, ∞)
17. Sketch the graph of the interval described by the inequality 5 < x ≤ 12.
(a) ✛ ❢ ✈
5 12
(b) ✛ ✈ ❢
5 12
(c) ✛ ✈ ✈
5 12
(d) ✛ ❢ ❢
5 12
✲
✲
✲
✲
Objective 7: Solve compound linear inequalities in one variable. Solve application problems
that require setting up and solving linear inequalities in one variable.
18. Solve the following compound inequality. Write your solution in interval notation.
(a) (−2, 1]
(b) (−1, 2]
(c) There is no solution.
(d) All real numbers are solutions.
−11 < 5x − 1 ≤ 4
5

19. Solve the following compound inequality. Write your solution in interval notation.
(a) (−∞, −1) ∪ [14, ∞)
(b) (−1, 14]
(c) (−2, 4]
(d) (−∞, −2) ∪ [4, ∞)
2x + 1 < −3 or x + 5 ≥ 9
20. Tess has $30 and she wants to go to the fair. It costs $10 for admission and $1.50
per ride. How many rides can she afford
(a) 11 rides
(b) 20 rides
(c) 13 rides
(d) 14 rides
Objective 8: Solve linear absolute-value equations.
21. Solve for p: |2p − 4| = 6
(a) p = 1
(b) p = 5 or p = −1
(c) p = −5 or p = 1
(d) p = 5 or p = 1
22. Solve for z:
3z + 5
∣ 6 ∣ − 3 = 6
(a) z = − 40
3 or z = −41 3
(b) z = − 39
3 or z = 39 3
(c) z = 4 or z = 5
(d) z = − 59
3 or z = 49 3
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23. Solve for m:
∣ ∣∣∣ ∣ −3 4 m + 8 + 7 = 5
(a) There is no solution.
(b) All real numbers are solutions.
(c) m = 8 or m = 40/3
(d) m = 40/3
Objective 9: Solve linear absolute-value inequalities. Graph solutions on the number line,
write solutions as inequalities, write solutions in interval notation, and write solutions
in set-builder notation.
24. Solve the following inequality: 7 + |5x − 3| ≥ 6
(a) x > 2/5
(b) x < −2 or x > 2
(c) There is no solution.
(d) All real numbers are solutions.
25. Solve the following inequality. Graph your solution on a number line.
1 − |x + 3| > −6
(a) ✛ ❢ ❢
−10 4
(b) ✛ ❢ ❢
−8 2
(c) ✛ ❢ ❢
−1 6
(d) ✛ ❢ ❢
−6 1
✲
✲
✲
✲
26. Solve the following inequality. Write your solution in interval notation.
(a) (−6, 2/3)
(b) (−∞, −6) ∪ (2/3, ∞)
(c) There is no solution.
(d) All real numbers are solutions.
|3x + 8| > 10
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Objective 10: Solve inequalities and systems of inequalities in two variables. Sketch the
graphs of the solution sets.
27. Which one of the graphs shown below illustrates the solution set of the following
system of inequalities
6x + 2y < 4
y + 3 ≥ 2x
(a)
y
(b)
y
8
8
6
6
4
4
2
-8 -6 -4 -2 2 4 6 8
-2
x
2
-8 -6 -4 -2 2 4 6 8
-2
x
-4
-4
-6
-6
-8
-8
(c)
y
(d)
y
8
8
6
6
4
4
2
-8 -6 -4 -2 2 4 6 8
-2
x
2
-8 -6 -4 -2 2 4 6 8
-2
x
-4
-4
-6
-6
-8
-8
8

28. Which one of the graphs shown below illustrates the solution set of the following
system of inequalities
x ≥ 0
y ≥ 0
x ≤ 7
(a)
y
(b)
y
8
8
6
6
4
4
2
-8 -6 -4 -2 2 4 6 8
-2
x
2
-8 -6 -4 -2 2 4 6 8
-2
x
-4
-4
-6
-6
-8
-8
(c)
y
(d)
y
8
8
6
6
4
4
2
-8 -6 -4 -2 2 4 6 8
-2
x
2
-8 -6 -4 -2 2 4 6 8
-2
x
-4
-4
-6
-6
-8
-8
9

Objective 11: Convert expressions involving rational exponents to radical expressions and
vice versa.
29. Rewrite with rational exponents: ( 5√ xy 2 ) 15
(a) (xy 2 ) 3/5
(b) (x 5 y 2 ) 1/3
(c) (xy 2 ) 5
(d) (xy 2 ) 3
30. Rewrite in radical form: (9x 3 y 2 ) 7/4
7
(a) √ 9xy 4
4
(b) √ 9x 3 y 2
(c) ( 4√ 9x 3 y 2 ) 7
(d) ( 7√ 9x 3 y 2 ) 4
Objective 12: Simplify radical expressions where the radicand is a monomial.
31. Assuming x and y represent positive numbers, simplify the following expression.
√
4x2 y 6
(a) 2xy 3
(b) 16x 4 y 12
(c) 2x 2 y 6
(d) 2x 4 y 3
32. Simplify:
3√
5a8 b 6 c 4
(a) 5a 2 b 2 c 3√ a 2 c 4
(b) a 2 b 2 c 3√ 5a 2 c 4
(c) a 2 b 2 c 3√ 5a 2 c
(d) 5a 2 b 2 c 3√ a 2 c
10

Objective 13: Add, subtract, and multiply radical expressions. Simplify the results. [The
expressions may consist of sums or differences of radicals. Only monomial radicands
are considered.]
33. Expand and simplify the following expression.
(a) − √ x
(b) x − 1
(c) x − √ x − 6
(d) 2 √ x − 1
(2 + √ x)( √ x − 3)
34. Expand and simplify the following expression.
(a) x + 9
(b) x + 3
(c) 6 + 2 √ x
(d) x + 6 √ x + 9
( √ x + 3) 2
35. Simplify and combine like radicals.
3√
x3 + 3√ x 3 y 2 + x 3√ y 2
(a) 3x + 2 3√ y 2
(b) x + 2x 3√ y 2
(c) 3x 3√ y 2
(d) x + 3√ xy 2 + x 3√ y 2
36. Simplify and combine like radicals.
3√
27 +
3 √ 81 − 4 3√ 3
(a) 3 − 3√ 3
(b) −2 3√ 111
(c) 2 3√ 3
(d) −2 3√ 3
11

Objective 14: Divide radical expressions and rationalize denominators. [Only monomial
radicands are considered.]
37. Rationalize the denominator and simplify:
2n
√
18n
(a) 1 9
(b)
(c)
√
2n
3
n
√
3
(d)
√
3n
2n
38. Rationalize the denominator and simplify:
(a) 2 + √ x
4 − √ x
2
2 − √ x
(b) 4 + 2√ x
4 − x
(c) 1 + 2√ x
1 − x
(d) 4 + 2√ x
4 − √ x
Objective 15: Solve radical equations where the radicand is a linear expression.
39. Solve for x:
√ 3x + 4 = −2
(a) x = 0
(b) x = −8/3
(c) x = 8/3
(d) There is no real solution.
40. Solve for y: 6 − 2 √ 3y = −10
(a) y = 64/3
(b) y = 8/3
(c) y = 16/3
(d) There is no real solution.
12

Objective 16: Simplify complex numbers and write in standard form a + bi. [Students
must be prepared to add, subtract, multiply, and divide complex numbers, as well as
write square roots of negative numbers in complex form.]
41. Expand and simplify. Write your result in standard form.
(a) 7i 2 + 2i − 2
(b) 12i 2 + 2i − 2
(c) −14 + 2i
(d) −7 + 2i
(2 + 4i)(3i − 1)
42. Expand and simplify. Write your result as a complex number in standard form.
√
−9(
√
−9 + 1)
(a) −9 + 3i
(b) 9 + 3i
(c) 9 + √ −9
(d) 10
43. Simplify and write in standard form:
(a) −3/2
(b) i
(c) 5i
(d) 2i2 + 5i + 2
−i 2 + 4
1 + 2i
2 − i
Objective 17: Solve all types of quadratic equations, including those with complex solutions.
Solve rational equations that reduce to quadratic equations.
x 2
44. Solve for x:
3 = −2 3 x − 5 3
(a) x = 1 + i or x = 1 − i
(b) x = −1 + 2i or x = −1 − 2i
(c) x = −2 + i or x = −2 − i
(d) x = 4 + 2i or x = 4 − 2i
13

45. Solve for y: 2y 2 + 5y − 3 = 0
(a) y = −2 or y = 1/2
(b) y = −3 or y = 2
(c) y = −1 or y = 1/2
(d) y = −3 or y = 1/2
46. Solve for x, and then find the sum of the two solutions.
(a) 2
(b) 2 √ 3
(c) 2 + √ 3
(d) 0
x − 2 = 2 x
Objective 18: Solve application problems that require setting up and solving quadratic
equations. [Students should be prepared to use simple geometric formulas such as
those giving the areas of rectangles and triangles.]
47. The product of two consecutive numbers is 56. If x is the greater of the two
numbers, then which quadratic equation represents the problem situation
(a) x(x + 1) = 56
(b) x(x − 1) = 56
(c) x + (x − 1) = 56
(d) x + (x + 1) = 56
48. The length of a room is 2ft less than twice its width. The area of the room is
60ft 2 . Find the length of the room.
(a) The length is 6ft.
(b) The length is 10ft.
(c) The length is 4ft.
(d) The length is 15ft.
49. A garden is twice as long as it is wide. The square of its perimeter is 4356ft 2 .
Find the width of the garden.
(a) The width is 8ft.
(b) The width is 16ft.
(c) The width is 22ft.
(d) The width is 11ft.
14

Objective 19: Graph quadratic functions. Find vertices and x- and y-intercepts of parabolas.
50. The function f is written below in two different, but equivalent, forms:
f(x) = 2(x − 2) 2 + 3 f(x) = 2x 2 − 8x + 11
Using either one of the forms for f, sketch the graph of f.
(a)
y
(b)
y
8
8
6
6
4
4
2
-8 -6 -4 -2 2 4 6 8
-2
x
2
-8 -6 -4 -2 2 4 6 8
-2
x
-4
-4
-6
-6
-8
-8
(c)
y
(d)
y
8
8
6
6
4
4
2
-8 -6 -4 -2 2 4 6 8
-2
x
2
-8 -6 -4 -2 2 4 6 8
-2
x
-4
-4
-6
-6
-8
-8
51. Which of the following is an equation for a parabola that opens up and has vertex
(−4, 5) (Each equation is written in two different, but equivalent, forms.)
(a) y = (x − 4) 2 + 5 or y = x 2 − 8x + 21
(b) y = (x + 4) 2 + 5 or y = x 2 + 8x + 21
(c) y = −(x + 4) 2 − 5 or y = −x 2 − 8x − 21
(d) y = −(x − 4) 2 − 5 or y = −x 2 + 8x − 21
15

52. Which quadratic function shown below has the given graph (Each function is
written in two different, but equivalent, forms.)
y
8
6
4
2
-8 -6 -4 -2 2 4 6 8
-2
-4
-6
-8
x
(a) f(x) = −2(x − 3) 2 + 2 or f(x) = −2x 2 + 12x − 16
(b) f(x) = −(x + 3) 2 + 2 or f(x) = −x 2 − 6x − 7
(c) f(x) = (x − 3) 2 − 2 or f(x) = x 2 − 6x + 7
(d) None of the above
Objective 20: Solve application problems involving the Pythagorean theorem.
53. The triangle shown below has the given dimensions, some of which depend on
the unknown quantity x. Use the Pythagorean theorem to set up and solve an
equation for x.
√
17
✘✘ ✘✘✘ ✘ ✘✘✘ ✘ ✘✘
x
(a) x = −4
(b) x = 1
(c) x = 7
(d) x = 4.86119
x + 3
54. The length of one leg of a right triangle is two inches more than twice the length
of the other leg. Find the perimeter of the triangle if the hypotenuse measures 13
inches.
(a) 33in
(b) 32in
(c) 31in
(d) 30in
16

Objective 21: Solve application problems involving direct and inverse variation.
55. When a car travels at a constant speed, the distance it travels varies directly as
the time of travel. If a car travels 87 miles in 2 hours, then how many miles does
it travel in 7 hours
(a) 609mi
(b) 304.5mi
(c) 407.3mi
(d) 24.86mi
56. The quantities x and y vary inversely so that x = 10 when y = 8. Find x when
y = 40.
(a) x = 50
(b) x = 100
(c) x = 5
(d) x = 2
Objective 22: Use substitution to solve polynomial equations that reduce to quadratic
equations.
57. Solve for x, and then find the sum of the two solutions.
(a) 4
(b) 1
(c) 8
(d) 3/16
4(2x + 1) 2 − 16(2x + 1) + 15 = 0
58. In order to solve for x, which is the most appropriate substitution
(a) let u = x 6
(b) let u = 2x 3
(c) let u = 3√ x
(d) let u = x 3 17
x 6 + 2x 3 = 8

Free Response Problems: On your final exam you must show all work to receive full
credit for the free response problems.
59. A medieval alchemist’s love potion calls for a number of eyes of newt and toes
of frog, the total being 21, but with twice as many newt eyes as frog toes. How
many of each are required Set up a system of linear equations that corresponds
to the problem situation and use any method to solve the system.
60. Graph the parabola given by the equation y = x 2 + 4x + 3. Find the coordinates
of the vertex and the x-intercepts.
✻
✲
18

61. Solve for x:
√ x + 3 = 2x.
62. Simplify:
5 √ 64x 8 y 15
63. Solve for x: |7 − 2x| − 7 = 15 + x
64. Solve and write your solution in interval notation.
2 < −8x + 4 ≤ 36
65. Determine the domain of the function f.
f(x) =
5
√ x − 2
19

Answer Key
1. c, Follow-up: Range is (−∞, 1] 21. b 41. c
2. a 22. d 42. a
3. c 23. a 43. b
4. a 24. d 44. b
5. a 25. a 45. d
6. c 26. b 46. a
7. d 27. a 47. b
8. b 28. a 48. b
9. b 29. d 49. d
10. d 30. c 50. d
11. c 31. a 51. b
12. b 32. c 52. a
13. d 33. c 53. b
14. a 34. d 54. d
15. b 35. b 55. b
16. c 36. a 56. d
17. a 37. b 57. b
18. a 38. b 58. d
19. d 39. d
20. c 40. a
For problems 59–65, see your instructor for help and evaluation of your work.
Prairie State College, Department of Mathematics
May 8, 2007
http://math.prairiestate.edu
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