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# College Algebra Study Guide

E.12

47. Solve the inequality x2 - x > 12.

48. Solve the inequality x3 - x2 ≤ 6x.

49. Solve the inequality .

51. A Web-based embroidery company makes monogrammed napkins. The pro t associated with
producing x orders of napkins is governed by the equation

P(x) = -x2 + 130x - 3000

Determine how many orders the company should accept to make a pro t.
Answer: 31 to 99 orders will yield a pro t. (i.e. (30, 100)).

E.13
52. State the order of the matrix 2

Answer: 3 x 2, (i.e 3 rows by 2 columns )

53. Solve for the indicated variables.

Answer: x = 6, y = 3
54. Let  and . Find 2A - 3B and BA.

55. The IRS allows and individual to deduct business expenses in the following way: \$0.45 per
mile driven, 50% of entertainment cost, and 100% of actual expenses. Represent these deductions as
a row matrix A. In 2006, Joe had the following business expenses: \$2, 700 in entertainment, \$15, 200
actual expenses, and he drove 7523 miles. Represent Joe's expenses as a column matrix B. Multiply
these matrices (AB) to find the total amount of business expenses Joe can claim on his 2006 tax form.

56. Find the inverse (if it exists) of the matrix

E.14

57. Find the fifth term in the expansion of (s - 3y)7.
Answer: The fifth term is 2835s3y4.

58. Apply the binomial theorem to expand (2z - 3y)4:
Answer: (2z - 3y)4 = 16z4 - 96z3y + 216z2y2 - 216zy3 + 81y4

E.15

59. The 54th and 4th term of an arithmetic sequence are -61 and 64, respectively. Find the 23rd
term in the sequence.

60. Find the sum of the first 11 terms of the Arithmetic sequence 9, 17, 25, 33, ...

61. Find the first four terms and the 10th term of the sequence given by .

62. Give the first four terms of the geometric sequence for which = -5 and r = 4:

63. Find the next three terms of the geometric sequence 27,-9, 3,-1, ...

64. For the sequence . write a formula for the general term an:

65. Find the infinite sum 0.5 + 0.05 + 0.005 +... .

66. Find the sum of the first seven terms of the sequence

E.16

67. Use Cramer's Rule to solve for x and y.

3x - 4y = 1
4x - 2y = 8

Answer: x = 3 and y = 2

68. Solve the system of equations by substitution or addition method:

x + y + z = 0
2x + z = -1
x - y - z = 2

x = 1, y = 2, z = -3

69. Solve the system by the method of your choice:

2u + 5v = 7
3u - v = 5

70. A Honda Accord gets approximately 26 mpg on the highway and 19 mpg in the city. You drove
349.5 miles on a full 16 gallon tank. Assuming you drove on both highway and city, approximately
how many miles did you drive in the city and how many on the highway?

Answer: 169 miles on the highway, 180.5 miles in the city .

71. Find the determinant

E. 17

72. Prove by mathematical induction that, for all positive integers n,

Solution:
STEP 1. First we show that the formula above holds true when n = 1, i.e.

STEP 2. Now we assume that the formula above holds true for any k < n, that is

STEP 3. If we can show that the formula above holds for k + 1 term, we are done. So, adding
k + 1 to both sides in step 2 we get,

note that the right hand side is the value when (k + 1) is substituted for n.
Hence if the formula is true for k, we have proved it is also true for k + 1, but since k is any
positive integer the formula is true for ALL positive integers n.

Extras:

73. Find the center and radius given the following equation of a circle:

x2 + y2 + 6x + 2y + 6 = 0