PMI Pre-Calculus Solving Equations
NJCTL.org
Basic Equations and Quadratics
Class Work
Solve the following equations or inequalities. Write the answers to inequalities in interval
notation.
1. -7(x – 3) + 4(x – 8) = 5(x – 2) – 3
2. 2(3x – 4) + 3(4x + 5) = 4 – (2x – 4)
3. 4(x – 7) – 2(3x + 5) > -12
4. -3(p – 4) – 2(p – 5) ≤ 5p + 6
5. 5a2 = 125
6. 6r3 + r2 – r = 0
7. 5x2 – 2x = 4
8. 2x2 + 11x + 12 > 0
9. x4 ≥ 4x3 – 5x2
10. 3m2 + 4 = 31
Basic Equations and Quadratics
Homework
Solve the following equations or inequalities. Write the answers to inequalities in interval
notation.
11. 6(m – 2) – (m – 7) = 3(m – 5) – 2
12. 4(2r – 4) + 3(r – 5) = 8 – (2r – 3)
13. -3(x – 3) + 3(2x + 6) ≤ -24
14. -5(g + 3) – (g – 5) > 3g – 2
15. x2 – x – 17 = -11
16. 5x2 – 13x + 6 > 0
17. 2x2 + x + 1 ≤ 0
18. 6x3 - 13x2 + 6x > 0
19. 2m2 +6 ≥ 38
20. 2x2 – 5x = -12x + 30
Spiral Review
Simplify:
21. √24𝑚3 𝑛6
Simplify:
22.
−12𝑥 −6 𝑦 3
15𝑥 −3 𝑦 −3
Factor:
23. 12x3 – 4x
Simplify:
24.
1
𝑥3
1
𝑥5
PMI Pre-Calculus Solving Equations
NJCTL.org
Absolute Value Equations
Class Work
Solve the following equations. Write your answers using interval notation where appropriate.
25. |2𝑥 − 6| = 12
26. −4|𝑥 − 3| = −12
27. −2|3𝑥 − 1| + 4 = −16
28. 14 = −2|3𝑥 − 8|
29. 4|𝑥 − 3| + 7 = 21
30. 4|2𝑥 − 8| ≥ 8
31. −3|𝑥 + 4| < −21
32. −5|𝑥 + 1| − 4 ≤ −17
33. 2|3𝑥 + 1| − 5 < 11
Absolute Value Equations
Homework
Solve the following equations. Write your answers using interval notation where appropriate.
34. −5|𝑥 − 3| = −5
35. −8|2𝑥 + 1| = 24
36. 3|2𝑥 − 1| − 5 = 22
37. 2|3𝑥 + 2| − 7 = 21
38. −2|𝑥 − 8| − 4 > 5
39. 5|2𝑥 + 3| − 8 ≤ 12
40. |−3𝑥 + 4| < 15
41. −|3𝑥 − 1| − 8 ≤ −15
42. −3|2𝑥 − 5| ≥ −8
Spiral Review
Factor:
Factor:
Simplify:
Simplify:
43. x3 – 27
44. 4m2 – 81
45. √36𝑟 5 𝑟 8
46.
4
𝑥
5
𝑏
PMI Pre-Calculus Solving Equations
NJCTL.org
Equations with Radicals
Class Work
Solve the following equations:
47. 4√𝑥 − 12 = 24
48. −√8 − 𝑥 + 2 = 12
49. 12√𝑚 − 4√𝑚 − 3 = 7
50. 8 − √3𝑥 + 1 = 2𝑥 + 1
52. √2𝑥 + 1 + 2 = 𝑥 − 1
51. 𝑟 = 2√𝑟 − 1
Equations with Radicals
Homework
Solve the following equations:
53. 2√𝑥 − 1 − 5 = 3𝑥
54. −√𝑥 + 3 = 3√𝑥 + 15
56. 5 − √2𝑥 + 2 = 𝑥 + 1
57. 5𝑚 = 2√𝑚 − 1
55. −√𝑝 + 4 − 4 = 7
58. 2√𝑥 + 1 − 3 = 𝑥 + 1
Spiral Review
Simplify:
59.
𝑎
𝑏
𝑐
𝑑
Work out:
Multiply:
Simplify:
60. (2x - 1)3
61. (-3r2s5)(-2r-2s-4)
62. √81𝑚6 𝑛4 𝑝16
PMI Pre-Calculus Solving Equations
NJCTL.org
Rational Equations
Class Work
Solve the following equations. Write your answers in interval notation where appropriate.
63.
𝑥
3
+ = +4
66.
3
𝑥
− 𝑥−2 = 𝑥
69.
72.
5
6
𝑥
4
6
𝑟 2 −9
𝑟 2 + 9𝑟 + 8
1
≤0
6𝑥 2 − 5𝑥 − 4
𝑥2− 4
<0
64.
3𝑥−1
𝑥+3
67.
𝑥
2
+ 𝑥 2 + 2𝑥 + 1
𝑥+1
70.
𝑚2 − 5𝑚 − 36
𝑚+3
73.
4𝑥 3 −16𝑥
𝑥−5
=
2
3
>0
≥0
𝑥
= 𝑥−5
65.
3
4
− 2
𝑥−3
𝑥 − 𝑥−6
68.
2
2
+ 𝑥2 − 4
𝑥2 + 𝑥 − 6
71.
𝑥(𝑥−3)(𝑥−4)
(𝑥+3)
74.
𝑥+1
𝑥 3 −9𝑥
≤0
≥0
=
5
𝑥+2
2
= − 𝑥+3
PMI Pre-Calculus Solving Equations
NJCTL.org
Rational Equations
Homework
Solve the following equations. Write your answers in interval notation where appropriate.
75.
4
3
𝑥
5
𝑥
6
76.
5𝑥 − 6
2𝑥 + 5
78.
2
3
+𝑥
𝑥−4
10
𝑥
79.
𝑥
3
+ 𝑥 2 + 5𝑥− 6
𝑥−1
81.
𝑟 2 −16
𝑟 2 + 𝑟− 12
≤0
82.
𝑚2 − 19𝑚 + 48
𝑚+ 5
84.
6𝑥 2 + 𝑥 − 1
𝑥 2 − 4𝑥 − 12
85.
𝑥 3 −4𝑥
𝑥
+ = −3
=
Spiral Review
Simplify:
87.
4𝑚
5
𝑚3
2
<0
=
3
5
≥0
𝑥
= 𝑥−5
>0
86.
77.
1
3
− 2
𝑥−5
𝑥 − 𝑥−6
80.
2
2
+ 𝑥2 − 9
𝑥−3
= − 𝑥+3
83.
𝑥(𝑥−2)(𝑥+6)
(𝑥−5)
≥0
𝑥+1
3𝑥 3 −27𝑥
=
2
≤0
Factor:
Multiply:
Simplify:
88. 27x3 + 8
89. (x2 – 2x + 1)(x2 + x – 6)
90. √48𝑟 5 𝑠 6
3
1
𝑥−3
PMI Pre-Calculus Solving Equations
NJCTL.org
Exponential and Logarithmic Equations
Class Work
91. 5𝑥 = 23
92. 62𝑑+5 = 79
93. 7𝑛−2 = 3𝑛
94. 3𝑡+2 = 4𝑡−2
95. 𝑙𝑜𝑔4 (𝑥 + 2) = 𝑙𝑜𝑔4 (3𝑥 − 12)
96. 𝑙𝑜𝑔3 5𝑥 + 𝑙𝑜𝑔3 8 = 15
97. 𝑙𝑜𝑔2 2𝑥 + 𝑙𝑜𝑔2 (𝑥 + 6) = 5
98. 𝑒 ln4 𝑥 = 24
99. 𝑒 ln 3𝑥 − 15 = 25
100. 5 ln 𝑒 3𝑥 + 12 = 7
Exponential and Logarithmic Equations
Homework
101. 3𝑥 = 21
102. 33 − 𝑛 = 3𝑛
105. 𝑙𝑜𝑔4 2𝑥 2 + 𝑙𝑜𝑔4 𝑥 = 𝑙𝑜𝑔4 54
104. 𝑙𝑜𝑔3 3𝑥 − 𝑙𝑜𝑔3 12 = 𝑙𝑜𝑔3 24
106. 𝑒 ln 7𝑥 = 21
107. 4𝑒 ln 𝑥 − 3 = 75
109. 3 ln 𝑒 2𝑥 − 12 = 35
Spiral Review
Simplify: (Be careful!)
111.
1
𝑥
1
+2
𝑥
103. 8𝑡+3 = 62𝑡−1
108. ln 𝑒 3𝑥 − 8 = 12 + 𝑥
110. 3𝑒 (2𝑥 − 1) + 12 = 10
Work out:
112.
3
∙
2
∙
5
16 15 4
Multiply:
Simplify:
113. (-4xy3z4)3
114. √64𝑚6 𝑛20
5
PMI Pre-Calculus Solving Equations
NJCTL.org
Multiple Choice
1. Solve: −2(𝑥 + 5) − 3(𝑥 + 4) = −(𝑥 + 5)
7
a. − 4
b. −
c.
d.
17
4
27
5
17
5
2. Solve: 3|6 − 𝑚| = 12
a. 2
b. 10
c. 2 and 10
d. no solution
3. Solve: |2𝑥 − 1| − 16 = 7
a. -11
b. 12
c. -11 and 12
d. no solution
4. Solve: −8(𝑝 + 12) ≤ −2(𝑝 − 3)
a. [−17, ∞)
b. (−∞. −17]
c. [−15, ∞)
d. (−∞, −15]
5. Solve: −6 ≤ 4 − 6𝑟 < 7
5
1
a. [ , − )
3
2
1 5
b. (− 2 , 3]
1
2
5
3
c. (−∞, − ) 𝑜𝑟 [ , ∞)
d. ∅
6. Given 3x = 12, find x:
a. 2.262
b. 1.386
c. 1.079
d. 0.477
7. Solve −5|3𝑥 − 4| ≥ −25
1
a. [− 2 , ∞)
b. (−∞, 3]
1
c. [− , 3]
2
d. no solution
PMI Pre-Calculus Solving Equations
8. Solve |7 − 3x| < −2
5
a. (3 , ∞)
b. (−∞, 3)
5
c. (3 , 3)
d. no solution
9. Solve |2𝑥 − 1| ≥ −3
a. [−1, ∞)
b. (−∞, −2)
c. (−∞, ∞)
d. no solution
3
1
10. (3)2 = (2x)2
a.
b.
c.
27
2
9
2
3
4
d. no solution
11. √2𝑥 + 3 = 𝑥 − 3
a. 4 ± √10
b. 2 and 4
c. 2
d. 4
12. 3𝑥 + 5 = √3𝑥 + 4
a. 9
b. 3 and 7
c. 21
d. no solution
13. Solve:
−2
5x − 6
=
6
2x + 3
a. 3
b.
c.
d.
7
5
15
17
15
13
4
2
5
14. Solve: x2 −x−61 + x−3 = x+2
a.
b.
23
3
7
3
c. -2 and 3
d. 6
NJCTL.org
PMI Pre-Calculus Solving Equations
2
3
4
15. Solve: x2 −9 + x2 −x−6 = x2 +5x+6
a.
b.
c.
d.
16. Solve:
-25
-8
8
no solution
2𝑥 + 5
3𝑥 − 9
<0
a. (3, ∞)
5
b. (−∞, − 2)
5
c. [− 2 , 3]
5
d. (− 2 , 3)
17.
𝑥 2 − 12𝑥 + 36
2𝑥 2 + 𝑥 − 15
≥0
5
a. (−∞, −3) ∪ (2 , ∞)
5
b. (−∞, −3) ∪ (2 , 6) ∪ (6, ∞)
5
2
c. (−3, )
d. (−∞, ∞)
18. log x x 7 = ?
a. 7x
b. 7
c. x/7
d. cannot be determined
19. Solve: 6𝑒 (2𝑥−1) + 11 = 23
a. -.153
b. .693
c. .847
d. 3.386
20. ln(𝑥 − 1) + 9 = 8
a. 0
b. e – 1
c. e-1 + 1
d. no solution
NJCTL.org
PMI Pre-Calculus Solving Equations
NJCTL.org
Extended Response
1. A tire company makes on average 3000 tires a day, give or take 50 tires.
a. Write a compound inequality to represent this situation.
b. Write an absolute value inequality to represent the situation.
c. Write a compound inequality to show how many chairs are made in 5 days.
2. | 3x - 4| ? y
a. What equality/inequality (?) and value of y should be used so that solution has 2
answers?
b. What equality/inequality (?) and value of y should be used so that solution has an
answer of all reals?
c. What equality/inequality (?) and value of y should be used so that solution has no
solution?
3. Using the inequality,
𝑥 2 + 2𝑥 + 5
6𝑥 2 + 19𝑥 + 10
≤ 0, answer the following questions.
a. Explain what you are finding when looking for the solution to the inequality.
b. Give two different ways to solve this inequality.
c. Which of the following numbers are solutions to the inequality?
5 2 2 5
−5, −3, − , − , , , 3,5
2 3 3 2
PMI Pre-Calculus Solving Equations
NJCTL.org
Answer Key
1
39. [-3.5,0.5]
1
40. (-33,63)
1. x=4
2
2. x=20
3. (-∞,-13)
4. [1.6,∞)
5. a=-5,5
6.
1
1
r=− 2,0,3
7. x=-0.716,1.116
8. (-∞,-4)∪(-1.5,∞)
9. (-∞,∞)
10. m=-3,3
11. m=-6
12.
42
r=13
13. (-∞,-17]
8
14. (-∞,− 9)
15. x=-2,3
3
5
16. (-∞, )∪(2,∞)
17. Never True
2
3
18. (0, )∪(1.5,∞)
19. (-∞,-4]∪[4,∞)
20. x=-1.5,10
21. 2𝑚𝑛3 √6𝑚
22.
−4𝑦 6
5𝑥 3
23. 4x(3x2-1)
24. x2
25. x=-3,9
26. x=0,6
27.
2
x=-3,3
3
28. No Solution
29. x=-0.5,6.5
30. (-∞,3]∪[5,∞)
31. (-∞,-11)∪(3,∞)
32. (-∞,-3.6)∪(1.6,∞)
1
33. (-3,23)
34. x=2,4
35. No Solution
36. x=-4,5
1
37. x=-4,53
38. Never True
1
2
3
41. (-∞,-2]∪[2 ,∞)
1
5
42. [16,36]
43. (x-3)(x2+3x+9x)
44. (2m+9)(2m-9)
45. 6𝑟 2 √𝑟
4𝑏
46. 5𝑥
47. x=81
48. No Solution
9
49. m=116
50. x=2.138
51. r=2
52. x=6.828
53. No Solution
54. No Solution
55. No Solution
56. x=1.683
57. No Solution
58. No Solution
𝑎𝑑
59. 𝑏𝑐
60. 8x3-12x2+6x-1
61. 6r4s
62. 9𝑚3 𝑛2 𝑝8
63. x=38
2
64. x=17
65. x=8.5
66. x=-1
67. x=
−2+2𝑖√14 −2−2𝑖√14
,
6
6
68. x=-1
69. (-8,-3]∪(-1,3]
70. (-4,-3)∪(9,∞)
71. (-∞,-3)∪[0,3}∪[4,∞)
1
1
72. (-2,− 2)∪(13,2)
73. (-∞,-2]∪[0,2]∪(5,∞)
74. (-3,-1]∪(0,3)
75. x=-130
76. x=2.368
77. x=19
PMI Pre-Calculus Solving Equations
3
98. x=6
78. x=55
1
79. x=-4.40,-0.85
80. No Solution
81. [-4,-3)
82. (-5,3)∪(16,∞)
83. (-∞,-6]∪[0,2]∪(5,∞)
1
2
1
3
84. (-2,− )∪( ,6)
85. (-∞,-2]∪[2,∞)
86. (-3,-1]∪(0,3)
8
87. 5𝑚2
88. (3x+2)(9x2-6x+4)
89. x4-x3-7x2+13x-6
3
90. 2𝑟𝑠 2 √6𝑟 2
91. x=1.95
92. d=-1.28
93.
94. t=17.28
95. x=7
96. x=6.075
97. x=2
Multiple Choice
1. B
2. C
3. C
4. A
5. B
6. A
7. C
8. D
9. C
10. A
11. A
12. D
13. C
14. A
15. A
16. D
17. B
18. B
19. C
20. C
NJCTL.org
99. x=133
100. x=−
101.
102.
103.
104.
105.
106.
107.
108.
1
3
x=2.77
x=1.5
x=5.34
x=96
x=3
x=3
x=19.5
x=2
5
109. x=76
110. No Solution
1
111. 2𝑥+1
112.
1
32
113. -64x3y9z12
114. 2𝑚𝑛4 5√2𝑚
Extended Response
4.
a. 2950 ≤ x ≤ 3050
b. |𝑥 − 3000| ≤ 50
c. 14750 ≤ x ≤ 15250
5.
a. Any positive number
b. y ≥ 0
c. y < 0
6.
a. When the fraction equals a
negative number or 0
b. Graph it, or make a
positive/negative table
c. None of the numbers are
solutions