Math 1010 Section 5
William Malone
Final Review
Name:
This review will be graded on completion and is due on December 14th. Complete
solutions can be found on my website.
1. Simplify the following using the order of operations
3 × 5 + 2 ÷ 6 − 4(−6 × 24 + 20) ÷ 2 × 6
2. Simplify the following
2
5
−
5
6
3
4
3. Simplify the following (4x − 1)(3x2 y − 6xy + y 2 ) − (5y + 2)(6x2 + 2x2 + xy).
4. Find all solutions to 3x2 − 4x(x − 2) + 4x − 3 = 5x2 − x(6x + 2) − 5
5. Graph all real numbers that satisfy both inequalities
3x + 2 > 4x + 1 and 6x − 2 ≥ 9 − x
6. Find all solutions to the equation |3x−1|−|x+2| = 10 and check your solutions.
7. Graph all solutions to the inequality on the real number line |2x − 5| > 6.
8. Given two points (3, 2) and (−4, 5) compute the distance between them and
find the midpoint on the line segment joining them.
9. Find the x and y-intercepts for the function f (x) = 12x2 − 13x − 14.
10. Find the equation for the line passing through (2, 3) of slope -13. Put your
answer in standard form.
11. Find the equation for the line passing through (4, −1) and (−5, 6). Put your
answer in slope intercept form.
12. Find an equation for the line passing through (−7, 1) perpendicular to the line
3x − 4y + 6 = 4y − 2x + 9.
13. Give an example of a line that is perpendicular to 4y-7x=10 and an example of
a line that is parallel to the same line.
14. Graph all solutions to 4x − 2y + 7 > 3x − 2y + 12 in the rectangular coordinate
system.
15. Graph the domain of the function on the real number line.
f (x) = log5 (x2 − 1) +
16. Let f (x) =
√
√
3x2 + 1 − 42 5 − x +
47
x2
x+2
+ 9x + 14
3x2 + 1 x > 0
. Find f (2), f (0), f (−5) and f (2x + 1) for x > 1.
x − 4x2 x ≤ 0
17. Using the point plotting method plot the graph of the function f (x) = |||x +
1| − 3| − 3|.
18. Is 3y 2 − 2x + 5 = 0 a function of x? Why or why not?
19. Is 6y 2 − 2y + 1 = 6y(x + 1 + y) − 2 a function of x? Why or why not?
20. Graph f (x) = x2 − 10x + 28 using shifts and reflections of known graphs.
21. Graph g(x) = −( 31 )x+2 − 7 using shifts and reflections of known graphs.
22. Graph h(x) = − log 1 (−x − 5) + 4 using shifts and reflections of known graphs.
2
23. Find all solutions to the following system of equations.
3x − 2y = 26
4x + 5y = −19
24. Graph all solutions to the system of inequalities in the rectangular coordinate
system.
3x + 4y − 8 > 0
5x − 6y < −6
3y < x + 12
25. Simplify the following expression
(310 x2 )−7 − (y 3 z 4 )5
(3−11 x2 y 3 z 4 )4
26. Multiply the following polynomials. (3x − 4x2 + x10 )(x5 − 7x3 + 1)
27. Factor completely 4x8 − 1000x3 + 500x5 − 8x6 .
28. Factor completely 15x2 + 11x − 60.
29. Find all solutions to the equation 4x3 − 17x2 + 10x − 11 = 3x3 − 24x2 + 13x + 10.
30. Simplify
x2 −3x+2
x2 −5x+4
x2 +3x+1
x2 −2x−8
31. Simplify the following to a rational expression.
x−3 x−5
−
x+2 x+9
32. Perform the division
3x3 − 2x + 5
x−2
33. Perform the division
3x3 − 5x2
x2 + 1
34. Find all solutions to the equation
7
4
−
=5
x+2 x−1
2
35. Simplify the following to an integer 64 3 .
1
1
36. Write the following as a radical of a single polynomial (x − 1) 2 (x + 1) 4 .
37. Simplify
√
√
3− 2
√
√
6− 2
38. Find all real solutions to the equation (4x − 1)14 = 11.
√
√
√
39. Find all solutions to the equation 2x − 3 − 5 − x = 5 − 1.
40. Simplify the following to a complex number
3−i
3+i
2−i
2+i
41. Let f (x) = 3x2 −3, g(x) = log9 (x2 +1), and h(x) = 3x+1 . Then find (f ◦g ◦h)(x)
and (g ◦ f ◦ h ◦ h)(x).
42. Let T (x) = −12 log2 (−2x2 −3)+17. Write T (x) as a composition of 4 functions.
43. (Will be on the test) Let f (x) = 5 − (3 − x)5 . Find f −1 (x).
1
44. Simplify the following to a rational number. log128 ( 256
).
45. Expand the following: log11 (x6 y 7 z 9 w10 ).
46. Condense the following to a logarithm of a single quantity.
4log2 (x) − 3log4 (y) − 5log16 (x + 1)
47. Find all solutions to the equation 3 · 42x−1 − 5 = 0.
48. Find all solutions to the equation 9 log 1 (x − 2) + 11 = 0
2
49. (Quadradic Formula) x=
50. u2 − v 2 =
51. u3 + v 3 =
52. u3 − v 3 =
53. loga (uv) =
54. loga ( uv ) =
55. loga (un ) =
56. (Change of Base Formula) loga (u) =
57. (Definition of a logarithm) loga (x) = y if and only if
58. (Distance Formula)
59. (Midpoint Formula)
60. (Slope)
61. (Point-Slope Equation for a Line)
62. (Slope-Intercept Equation for a Line)
63. (General Equation for a Line)