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Appendix B Answers to Exercises
1 Linear 1.1 Modeling with Equations and Inequalities 1.1.5 Exercises
Review and Warmup 1.1.5.1. 1.1.5.1.a
Answer 1 .
\(a\hbox{, }A\hbox{, }x\hbox{, }y\hbox{, or }z\)
Answer 2 . 1.1.5.1.b
Answer 1 .
\(a\hbox{, }A\hbox{, }x\hbox{, }y\hbox{, or }z\)
Answer 2 .
\({\text{years or months}}\)
1.1.5.1.c
Answer 1 .
\(t\hbox{, }T\hbox{, }x\hbox{, }y\hbox{, or }z\)
Answer 2 .
\({\text{hours or minutes}}\)
Applications
1.1.5.3. 1.1.5.5. 1.1.5.7. 1.1.5.9. 1.1.5.11. 1.1.5.13. 1.1.5.15. 1.1.5.17. 1.1.5.19. 1.1.5.21. 1.1.5.23. 1.1.5.25. 1.1.5.27.
1.1.5.29. 1.1.5.31. 1.1.5.33. 1.1.5.35. Challenge 1.1.5.37.
1.2 Linear Equations and Inequalities 1.2.13 Exercises
Review and Warmup
1.2.13.1. 1.2.13.3. 1.2.13.5. Answer .
\(\left\{\frac{9}{7}\right\}\)
1.2.13.7.
Answer 1 .
\(\left\{\text{interval}, \left(-\infty ,-4\right]\right\}\)
Answer 2 .
\(\left(-\infty ,-4\right]\)
Answer 3 .
\(\left\{Y \mid Y\le -4\right\}\)
1.2.13.9.
Answer 1 .
\(\left\{\text{interval}, \left(-8,\infty \right)\right\}\)
Answer 2 .
\(\left(-8,\infty \right)\)
Answer 3 .
\(\left\{j \mid j > -8\right\}\)
1.2.13.11. 1.2.13.13. 1.2.13.15. Notation 1.2.13.17. Answer .
\(\text{no real solutions}\)
Skills Practice
1.2.13.19. 1.2.13.21. 1.2.13.23. 1.2.13.25. Answer .
\(\left\{\frac{4}{9}\right\}\)
1.2.13.27. Answer .
\(\left\{\frac{3}{5}\right\}\)
1.2.13.29. Answer .
\(\left\{\frac{1}{2}\right\}\)
1.2.13.31. Answer .
\(\left\{0.0704225\right\}\)
1.2.13.33. Answer .
\(\left\{-1.13559\right\}\)
1.2.13.35.
1.2.13.37. 1.2.13.39. 1.2.13.41. 1.2.13.43. 1.2.13.45. 1.2.13.47. 1.2.13.49. Answer .
\(\left\{\frac{-12}{13}\right\}\)
1.2.13.51. Answer .
\(\left\{\frac{-1}{4}\right\}\)
1.2.13.53. Answer .
\(\left\{\frac{14}{19}\right\}\)
1.2.13.55. Answer .
\(\left\{\frac{31}{2}\right\}\)
1.2.13.57. 1.2.13.59. 1.2.13.61. Answer .
\(\left\{\frac{-1}{2}\right\}\)
1.2.13.63.
Answer 1 .
\(\left\{\text{interval}, \left(3,\infty \right)\right\}\)
Answer 2 .
\(\left(3,\infty \right)\)
Answer 3 .
\(\left\{i \mid i > 3\right\}\)
1.2.13.65.
Answer 1 .
\(\left\{\text{interval}, \left(-6,\infty \right)\right\}\)
Answer 2 .
\(\left(-6,\infty \right)\)
Answer 3 .
\(\left\{u \mid u > -6\right\}\)
1.2.13.67.
Answer 1 .
\(\left\{\text{interval}, \left(-\infty ,-2\right)\right\}\)
Answer 2 .
\(\left(-\infty ,-2\right)\)
Answer 3 .
\(\left\{F \mid F < -2\right\}\)
1.2.13.69.
Answer 1 .
\(\left\{\text{interval}, \left(-\infty ,2\right)\right\}\)
Answer 2 .
\(\left(-\infty ,2\right)\)
Answer 3 .
\(\left\{R \mid R < 2\right\}\)
1.2.13.71.
Answer 1 .
\(\left\{\text{interval}, \left(-\infty ,6\right)\right\}\)
Answer 2 .
\(\left(-\infty ,6\right)\)
Answer 3 .
\(\left\{c \mid c < 6\right\}\)
1.2.13.73.
Answer 1 .
\(\left\{\text{interval}, \left[-3,\infty \right)\right\}\)
Answer 2 .
\(\left[-3,\infty \right)\)
Answer 3 .
\(\left\{n \mid n\ge -3\right\}\)
1.2.13.75.
Answer 1 .
\(\left\{\text{interval}, \left(1,\infty \right)\right\}\)
Answer 2 .
\(\left(1,\infty \right)\)
Answer 3 .
\(\left\{z \mid z > 1\right\}\)
1.2.13.77.
Answer 1 .
\(\left\{\text{interval}, \left(5,\infty \right)\right\}\)
Answer 2 .
\(\left(5,\infty \right)\)
Answer 3 .
\(\left\{K \mid K > 5\right\}\)
1.2.13.79.
Answer 1 .
\(\left\{\text{interval}, \left(-\infty ,-4\right)\right\}\)
Answer 2 .
\(\left(-\infty ,-4\right)\)
Answer 3 .
\(\left\{X \mid X < -4\right\}\)
1.2.13.81.
Answer 1 .
\(\left\{\text{interval}, \left(-\infty ,5\right]\right\}\)
Answer 2 .
\(\left(-\infty ,5\right]\)
Answer 3 .
\(\left\{i \mid i\le 5\right\}\)
1.2.13.83.
Answer 1 .
\(\left\{\text{interval}, \left(-\infty ,4\right]\right\}\)
Answer 2 .
\(\left(-\infty ,4\right]\)
Answer 3 .
\(\left\{t \mid t\le 4\right\}\)
1.2.13.85.
Answer 1 .
\(\left\{\text{interval}, \left[-5,\infty \right)\right\}\)
Answer 2 .
\(\left[-5,\infty \right)\)
Answer 3 .
\(\left\{F \mid F\ge -5\right\}\)
1.2.13.87.
Answer 1 .
\(\left\{\text{interval}, \left(-\infty ,-1\right)\right\}\)
Answer 2 .
\(\left(-\infty ,-1\right)\)
Answer 3 .
\(\left\{R \mid R < -1\right\}\)
1.2.13.89.
Answer 1 .
\(\left\{\text{interval}, \left(-\infty ,3\right]\right\}\)
Answer 2 .
\(\left(-\infty ,3\right]\)
Answer 3 .
\(\left\{c \mid c\le 3\right\}\)
1.2.13.91.
Answer 1 . Answer 2 .
\(\left\{\frac{9}{25}\right\}\)
1.2.13.93.
Answer 1 . Answer 2 .
\(\left\{\frac{-2}{5}\right\}\)
1.2.13.95.
Answer 1 . Answer 2 .
\(\left\{\frac{-37}{7}\right\}\)
1.2.13.97.
Answer 1 . Answer 2 .
\(\left\{\frac{248}{35}\right\}\)
1.2.13.99.
Answer 1 . Answer 2 .
\(\left\{\frac{-176}{25}\right\}\)
1.2.13.101. 1.2.13.103.
Answer 1 . Answer 2 .
\(\left\{\frac{-85}{97}\right\}\)
1.2.13.105.
Answer 1 . Answer 2 .
\(\left\{\frac{44}{147}\right\}\)
1.2.13.107.
Answer 1 . Answer 2 .
\(\left\{\frac{-11}{312}\right\}\)
1.2.13.109.
1.2.13.111.
Answer 1 . Answer 2 .
\(\left\{x \mid x < \frac{3}{28}\right\}\)
Answer 3 .
\(\left(-\infty ,{\frac{3}{28}}\right)\)
1.2.13.113.
Answer 1 . Answer 2 .
\(\left\{x \mid x > \frac{37}{2}\right\}\)
Answer 3 .
\(\left({\frac{37}{2}},\infty \right)\)
1.2.13.115.
Answer 1 . Answer 2 .
\(\left\{x \mid x < \frac{-408}{5}\right\}\)
Answer 3 .
\(\left(-\infty ,-{\frac{408}{5}}\right)\)
1.2.13.117.
Answer 1 . Answer 2 .
\(\left\{x \mid x\le -1\right\}\)
Answer 3 .
\(\left(-\infty ,-1\right]\)
1.2.13.119.
Answer 1 . Answer 2 .
\(\left\{x \mid x < \frac{-10}{33}\right\}\)
Answer 3 .
\(\left(-\infty ,-{\frac{10}{33}}\right)\)
1.2.13.121. Answer .
\(\text{no real solutions}\)
1.2.13.123. Answer .
\(\left(-\infty ,\infty \right)\hbox{ or }\text{infinitely many solutions}\)
1.2.13.125. Answer .
\(\left(-\infty ,\infty \right)\hbox{ or }\text{infinitely many solutions}\)
1.2.13.127. Answer .
\(\text{no real solutions}\)
1.2.13.129. Answer .
\(\text{no real solutions}\)
1.2.13.131. Answer .
\(\text{no real solutions}\)
1.2.13.133. Answer .
\(\left(-\infty ,\infty \right)\hbox{ or }\text{infinitely many solutions}\)
1.2.13.135. Answer .
\(\text{no real solutions}\)
1.2.13.137. Answer .
\(\left(-\infty ,\infty \right)\hbox{ or }\text{infinitely many solutions}\)
1.2.13.139. Answer .
\(\left(-\infty ,\infty \right)\hbox{ or }\text{infinitely many solutions}\)
1.2.13.141. Answer .
\(\left(-\infty ,\infty \right)\hbox{ or }\text{infinitely many solutions}\)
1.2.13.143. Answer .
\(\text{no real solutions}\)
1.2.13.145. Answer .
\(\left(-\infty ,\infty \right)\hbox{ or }\text{infinitely many solutions}\)
1.2.13.147. Answer .
\(\text{no real solutions}\)
1.2.13.149. Answer .
\(\text{no real solutions}\)
1.2.13.151. Answer .
\(\left(-\infty ,\infty \right)\hbox{ or }\text{infinitely many solutions}\)
1.2.13.153. Answer .
\(\left(-\infty ,\infty \right)\hbox{ or }\text{infinitely many solutions}\)
1.2.13.155. Answer .
\(\text{no real solutions}\)
1.2.13.157. Answer .
\(\text{no real solutions}\)
1.2.13.159. 1.2.13.159.a 1.2.13.159.b 1.2.13.161. 1.2.13.161.a 1.2.13.161.b 1.2.13.163. 1.2.13.163.a Answer .
\(\left\{\frac{2}{5}\right\}\)
1.2.13.163.b 1.2.13.165. 1.2.13.165.a 1.2.13.165.b 1.2.13.167. 1.2.13.167.a Answer .
\(\left\{\frac{-4}{9}\right\}\)
1.2.13.167.b
1.2.13.169. 1.2.13.171. 1.2.13.173. 1.2.13.175. Answer .
\(D = Q\mathopen{}\left(i-c\right)\)
1.2.13.177. 1.2.13.179. Answer .
\(y = -{\frac{7}{2}}x+13\)
1.2.13.181. 1.2.13.183. Answer .
\(y = {\frac{12}{13}}x+-{\frac{287}{52}}\)
Applications 1.2.13.185. 1.2.13.187. 1.2.13.189. 1.2.13.191. 1.2.13.193. 1.2.13.195. 1.2.13.197. 1.2.13.199. 1.2.13.201. 1.2.13.203. 1.2.13.205. 1.2.13.205.a 1.2.13.205.b 1.2.13.205.c Answer .
\(\left(0,304.348\right]\)
1.2.13.207. 1.2.13.207.a 1.2.13.207.b Answer .
\(10.9091\ {\textstyle\frac{\rm\mathstrut min}{\rm\mathstrut mi}}\)
1.2.13.207.c Answer .
\(\left(0,10.9091\right)\hbox{ or }\left(-\infty ,10.9091\right)\)
1.2.13.209. 1.2.13.209.a 1.2.13.209.b 1.2.13.209.c Answer .
\(\left[50,91.6667\right]\)
1.2.13.211. 1.2.13.213. Answer .
\(\frac{25}{4}\ {\rm min}\)
1.2.13.215. 1.2.13.217. 1.2.13.219. 1.2.13.221. 1.2.13.229. Answer .
\(h = \frac{A-2\pi r^{2}}{2\pi r}\)
Challenge 1.2.13.231.
1.3 Slope 1.3.16 Exercises
Skills Practice
1.3.16.1. 1.3.16.3. 1.3.16.5. 1.3.16.7. 1.3.16.9. 1.3.16.11. 1.3.16.13. 1.3.16.15.
1.3.16.17. 1.3.16.19. 1.3.16.21. 1.3.16.23. 1.3.16.25. 1.3.16.27.
1.3.16.29. 1.3.16.31. 1.3.16.33. 1.3.16.35. 1.3.16.37. 1.3.16.39. 1.3.16.41. 1.3.16.43. 1.3.16.45.
1.3.16.47. 1.3.16.49. 1.3.16.51. 1.3.16.53. 1.3.16.55. 1.3.16.57.
1.3.16.59. 1.3.16.61. 1.3.16.63.
1.3.16.65. Answer .
\(\left\{\text{line}, \text{solid}, \left(-2,-1\right), \left(3,2\right)\right\}\)
1.3.16.67. Answer .
\(\left\{\text{line}, \text{solid}, \left(2,-2\right), \left(7,-5\right)\right\}\)
1.3.16.69. Answer .
\(\left\{\text{line}, \text{solid}, \left(-8,-8\right), \left(-5,-7\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,-8\right), \left(-4,-7\right)\right\}, \left\{\text{line}, \text{solid}, \left(-6,-8\right), \left(-3,-7\right)\right\}, \left\{\text{line}, \text{solid}, \left(-5,-8\right), \left(-2,-7\right)\right\}, \left\{\text{line}, \text{solid}, \left(-4,-8\right), \left(-1,-7\right)\right\}, \left\{\text{line}, \text{solid}, \left(-3,-8\right), \left(0,-7\right)\right\}, \left\{\text{line}, \text{solid}, \left(-2,-8\right), \left(1,-7\right)\right\}, \left\{\text{line}, \text{solid}, \left(-1,-8\right), \left(2,-7\right)\right\}, \left\{\text{line}, \text{solid}, \left(0,-8\right), \left(3,-7\right)\right\}, \left\{\text{line}, \text{solid}, \left(1,-8\right), \left(4,-7\right)\right\}, \left\{\text{line}, \text{solid}, \left(2,-8\right), \left(5,-7\right)\right\}, \left\{\text{line}, \text{solid}, \left(3,-8\right), \left(6,-7\right)\right\}, \left\{\text{line}, \text{solid}, \left(4,-8\right), \left(7,-7\right)\right\}, \left\{\text{line}, \text{solid}, \left(5,-8\right), \left(8,-7\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,-7\right), \left(-5,-6\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,-6\right), \left(-5,-5\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,-5\right), \left(-5,-4\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,-4\right), \left(-5,-3\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,-3\right), \left(-5,-2\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,-2\right), \left(-5,-1\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,-1\right), \left(-5,0\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,0\right), \left(-5,1\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,1\right), \left(-5,2\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,2\right), \left(-5,3\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,3\right), \left(-5,4\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,4\right), \left(-5,5\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,5\right), \left(-5,6\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,6\right), \left(-5,7\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,7\right), \left(-5,8\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,-7\right), \left(-4,-6\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,-6\right), \left(-4,-5\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,-5\right), \left(-4,-4\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,-4\right), \left(-4,-3\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,-3\right), \left(-4,-2\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,-2\right), \left(-4,-1\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,-1\right), \left(-4,0\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,0\right), \left(-4,1\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,1\right), \left(-4,2\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,2\right), \left(-4,3\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,3\right), \left(-4,4\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,4\right), \left(-4,5\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,5\right), \left(-4,6\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,6\right), \left(-4,7\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,7\right), \left(-4,8\right)\right\}, \left\{\text{line}, \text{solid}, \left(-6,-7\right), \left(-3,-6\right)\right\}, \left\{\text{line}, \text{solid}, \left(-6,-6\right), \left(-3,-5\right)\right\}, \left\{\text{line}, \text{solid}, \left(-6,-5\right), \left(-3,-4\right)\right\}, \left\{\text{line}, \text{solid}, \left(-6,-4\right), \left(-3,-3\right)\right\}, \left\{\text{line}, \text{solid}, \left(-6,-3\right), \left(-3,-2\right)\right\}, \left\{\text{line}, \text{solid}, \left(-6,-2\right), \left(-3,-1\right)\right\}, \left\{\text{line}, \text{solid}, \left(-6,-1\right), \left(-3,0\right)\right\}, \left\{\text{line}, \text{solid}, \left(-6,0\right), \left(-3,1\right)\right\}, \left\{\text{line}, \text{solid}, \left(-6,1\right), \left(-3,2\right)\right\}, \left\{\text{line}, \text{solid}, \left(-6,2\right), \left(-3,3\right)\right\}, \left\{\text{line}, \text{solid}, \left(-6,3\right), \left(-3,4\right)\right\}, \left\{\text{line}, \text{solid}, \left(-6,4\right), \left(-3,5\right)\right\}, \left\{\text{line}, \text{solid}, \left(-6,5\right), \left(-3,6\right)\right\}, \left\{\text{line}, \text{solid}, \left(-6,6\right), \left(-3,7\right)\right\}, \left\{\text{line}, \text{solid}, \left(-6,7\right), \left(-3,8\right)\right\}\)
1.3.16.71. Answer .
\(\left\{\text{line}, \text{solid}, \left(-8,-13\right), \left(-6,-18\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,-12\right), \left(-6,-17\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,-11\right), \left(-6,-16\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,-10\right), \left(-6,-15\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,-9\right), \left(-6,-14\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,-8\right), \left(-6,-13\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,-7\right), \left(-6,-12\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,-6\right), \left(-6,-11\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,-5\right), \left(-6,-10\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,-4\right), \left(-6,-9\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,-3\right), \left(-6,-8\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,-2\right), \left(-6,-7\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,-1\right), \left(-6,-6\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,0\right), \left(-6,-5\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,1\right), \left(-6,-4\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,2\right), \left(-6,-3\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,3\right), \left(-6,-2\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,4\right), \left(-6,-1\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,5\right), \left(-6,0\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,6\right), \left(-6,1\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,7\right), \left(-6,2\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,8\right), \left(-6,3\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,9\right), \left(-6,4\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,10\right), \left(-6,5\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,11\right), \left(-6,6\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,12\right), \left(-6,7\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,13\right), \left(-6,8\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,-13\right), \left(-5,-18\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,-12\right), \left(-5,-17\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,-11\right), \left(-5,-16\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,-10\right), \left(-5,-15\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,-9\right), \left(-5,-14\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,-8\right), \left(-5,-13\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,-7\right), \left(-5,-12\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,-6\right), \left(-5,-11\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,-5\right), \left(-5,-10\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,-4\right), \left(-5,-9\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,-3\right), \left(-5,-8\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,-2\right), \left(-5,-7\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,-1\right), \left(-5,-6\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,0\right), \left(-5,-5\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,1\right), \left(-5,-4\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,2\right), \left(-5,-3\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,3\right), \left(-5,-2\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,4\right), \left(-5,-1\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,5\right), \left(-5,0\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,6\right), \left(-5,1\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,7\right), \left(-5,2\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,8\right), \left(-5,3\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,9\right), \left(-5,4\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,10\right), \left(-5,5\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,11\right), \left(-5,6\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,12\right), \left(-5,7\right)\right\}, \left\{\text{line}, \text{solid}, \left(-7,13\right), \left(-5,8\right)\right\}\)
1.3.16.73. 1.3.16.75. 1.3.16.77. 1.3.16.79. 1.3.16.81. 1.3.16.83. 1.3.16.85. 1.3.16.87.
1.3.16.89. 1.3.16.91. Answer .
\(y = \left({\frac{1}{4}}\right)x-1\)
1.3.16.93. 1.3.16.95. 1.3.16.97. Answer .
\(y = \left({\frac{1}{3}}\right)x-3\)
1.3.16.99. Answer .
\(y = -\left({\frac{3}{5}}\right)x\)
1.3.16.101. Answer .
\(\left\{\text{line}, \text{solid}, \left(0,-5\right), \left(1,-1\right)\right\}\)
1.3.16.103. Answer .
\(\left\{\text{line}, \text{solid}, \left(0,2\right), \left(1,-3\right)\right\}\)
1.3.16.105. Answer .
\(\left\{\text{line}, \text{solid}, \left(0,4\right), \left(-2,5\right)\right\}\)
1.3.16.107. Answer .
\(\left\{\text{line}, \text{solid}, \left(0,5\right), \left(3,6\right)\right\}\)
1.3.16.109. Answer .
\(\left\{\text{line}, \text{solid}, \left(0,0\right), \left(9,5\right)\right\}\)
1.3.16.111. Answer .
\(\left\{\text{line}, \text{solid}, \left(0,-1\right), \left(8,6\right)\right\}\)
1.3.16.113. Answer .
\(\left\{\text{line}, \text{solid}, \left(0,8\right), \left(1,9\right)\right\}\)
1.3.16.115. Answer .
\(\left\{\text{line}, \text{solid}, \left(0,-6\right), \left(1,-7\right)\right\}\)
1.3.16.117. Answer .
\(\left\{\text{line}, \text{solid}, \left(0,-2\right), \left(3,-7\right)\right\}\)
1.3.16.119. Answer .
\(\left\{\text{line}, \text{solid}, \left(0,2\right), \left(10,4\right)\right\}\)
1.3.16.121. 1.3.16.123. 1.3.16.125. 1.3.16.127. Answer .
\(y = \left({\frac{5}{2}}\right)x+6\)
1.3.16.129. Answer .
\(y = -\left({\frac{3}{4}}\right)x+4\)
1.3.16.131. Answer .
\(y = \left({\frac{14}{19}}\right)x+96\)
1.3.16.133.
1.3.16.135. 1.3.16.137. 1.3.16.139. 1.3.16.141.
1.3.16.143. Answer .
\(y = 4\mathopen{}\left(x-3\right)+2\)
1.3.16.145. Answer .
\(y = 5\mathopen{}\left(x+7\right)-9\)
1.3.16.147. Answer .
\(y = 4.8\mathopen{}\left(x+7\right)+3\)
1.3.16.149. Answer .
\(y = 9\mathopen{}\left(x-3\right)+6\hbox{ or }y = 9\mathopen{}\left(x-7\right)+42\)
1.3.16.151. Answer .
\(y = -7\mathopen{}\left(x-2\right)-3\hbox{ or }y = -7\mathopen{}\left(x+6\right)+53\)
1.3.16.153. Answer .
\(y = -\frac{9}{4}\mathopen{}\left(x+3\right)+6\hbox{ or }y = -\frac{9}{4}\mathopen{}\left(x-5\right)-12\)
1.3.16.155. Answer .
\(y = \frac{1}{3}\mathopen{}\left(x-2\right)-8\hbox{ or }y = \frac{1}{3}\mathopen{}\left(x+25\right)-17\)
1.3.16.157. 1.3.16.159. 1.3.16.161. Answer .
\(y = \frac{5}{6}x-\frac{4}{3}\)
1.3.16.163. Answer .
\(y = 4.5x+\left(-2.55\right)\)
1.3.16.165. Answer .
\(y = \frac{4}{5}\mathopen{}\left(x+3\right)-5\hbox{, }y = \frac{4}{5}\mathopen{}\left(x-2\right)-1\hbox{, or }y = \frac{4}{5}\mathopen{}\left(x-7\right)+3\)
1.3.16.167. Answer .
\(y = -\left(\frac{2}{5}\mathopen{}\left(x+5\right)+1\right)\hbox{, }y = -\left(\frac{2}{5}x+3\right)\hbox{, or }y = -\left(\frac{2}{5}\mathopen{}\left(x-5\right)+5\right)\)
1.3.16.169. Answer .
\(y = 20\mathopen{}\left(x-4\right)\hbox{ or }y = 20\mathopen{}\left(x-9\right)+100\)
1.3.16.171. Answer .
\(y = 348-11\mathopen{}\left(x-2\right)\hbox{ or }y = 304-11\mathopen{}\left(x-6\right)\)
1.3.16.173. Answer .
\(\left\{\text{line}, \text{solid}, \left(4,5\right), \left(5,8\right)\right\}\)
1.3.16.175. Answer .
\(\left\{\text{line}, \text{solid}, \left(5,-2\right), \left(6,1\right)\right\}\)
1.3.16.177. Answer .
\(\left\{\text{line}, \text{solid}, \left(-5,-3\right), \left(-2,-2\right)\right\}\)
1.3.16.179. Answer .
\(\left\{\text{line}, \text{solid}, \left(8,-2\right), \left(15,-11\right)\right\}\)
1.3.16.181. Answer .
\(\left\{\text{line}, \text{solid}, \left(9,-7\right), \left(19,-15\right)\right\}\)
Applications 1.3.16.183. 1.3.16.183.a Answer .
\(7559\ {\textstyle\frac{\rm\mathstrut people}{\rm\mathstrut yr}}\)
1.3.16.183.b Answer .
\(11120\ {\textstyle\frac{\rm\mathstrut people}{\rm\mathstrut yr}}\)
1.3.16.183.c Answer .
\(\left[2003\right], \left[2019\right], \left[2020\right], \left[2021\right]\)
1.3.16.183.d Answer .
\(\left[2006,2016\right]\)
1.3.16.183.e Answer .
\(10203.9\ {\textstyle\frac{\rm\mathstrut people}{\rm\mathstrut yr}}\)
1.3.16.185.
Answer 1 .
\(375\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut h}}\)
Answer 2 .
\(125\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut h}}\)
Answer 3 .
\(0\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut h}}\)
Answer 4 .
\(-500\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut h}}\)
1.3.16.187. Answer .
\(-3.33333\ {\textstyle\frac{\rm\mathstrut ml}{\rm\mathstrut min}}\)
1.3.16.189.
Answer 1 .
\(y = 0.04\mathopen{}\left(x-230\right)+22.2\)
Answer 2 . Answer 3 . 1.3.16.191.
Answer 1 .
\(y = 0.22\mathopen{}\left(x-10\right)+17.9\)
Answer 2 . Answer 3 . 1.3.16.193.
Answer 1 .
\(y = -44000\mathopen{}\left(x-4\right)+780000\hbox{ or }y = -44000\mathopen{}\left(x-6\right)+692000\)
Answer 2 . Answer 3 . 1.3.16.195.
Answer 1 .
\(y = -2.8\mathopen{}\left(x-9\right)+86.8\hbox{ or }y = -2.8\mathopen{}\left(x-19\right)+58.8\)
Answer 2 . Answer 3 . Challenge 1.3.16.197. 1.3.16.199. 1.3.16.201. 1.3.16.203. 1.3.16.205.
Answer 1 . Answer 2 . Answer 3 . 1.3.16.207.
Answer 1 . Answer 2 . Answer 3 . 1.3.16.209.
Answer 1 . Answer 2 . Answer 3 . 1.3.16.211.
Answer 1 . Answer 2 . Answer 3 .
1.4 Linear Systems 1.4.8 Exercises
Skills Practice
1.4.8.1. Answer .
\(\text{Yes, it is a solution.}\)
1.4.8.3. Answer .
\(\text{No, it is not a solution.}\)
1.4.8.5. Answer .
\(\text{No, it is not a solution.}\)
1.4.8.7. 1.4.8.9. 1.4.8.11.
1.4.8.13.
Answer 1 . Answer 2 .
\(\text{None of the above}\)
1.4.8.15.
Answer 1 .
\(\text{Infinitely many}\)
Answer 2 . 1.4.8.17.
1.4.8.19.
Answer 1 .
\(\left\{\text{line}, \text{solid}, \left(0,8\right), \left(7,20\right)\right\}, \left\{\text{line}, \text{solid}, \left(0,-6\right), \left(7,-8\right)\right\}\)
Answer 2 . 1.4.8.21.
Answer 1 .
\(\left\{\text{line}, \text{solid}, \left(5,4\right), \left(13,7\right)\right\}, \left\{\text{line}, \text{solid}, \left(3,-4\right), \left(9,-9\right)\right\}\)
Answer 2 . 1.4.8.23.
Answer 1 .
\(\left\{\text{line}, \text{solid}, \left(2,2\right), \left(3,-1\right)\right\}, \left\{\text{line}, \text{solid}, \left(-8,-4\right), \left(-7,-7\right)\right\}\)
Answer 2 . 1.4.8.25.
Answer 1 .
\(\left\{\text{line}, \text{solid}, \left(3,0\right), \left(0,-4\right)\right\}, \left\{\text{line}, \text{solid}, \left(4,0\right), \left(0,-8\right)\right\}\)
Answer 2 . 1.4.8.27.
Answer 1 .
\(\left\{\text{line}, \text{solid}, \left(-5,0\right), \left(0,-1\right)\right\}, \left\{\text{line}, \text{solid}, \left(5,-2\right), \left(0,-9\right)\right\}\)
Answer 2 . 1.4.8.29.
Answer 1 .
\(\left\{\text{line}, \text{solid}, \left(-3,0\right), \left(0,3\right)\right\}, \left\{\text{line}, \text{solid}, \left(-3,-8\right), \left(0,-5\right)\right\}\)
Answer 2 . 1.4.8.31.
Answer 1 .
\(\left\{\text{line}, \text{solid}, \left(-1,0\right), \left(0,-5\right)\right\}, \left\{\text{line}, \text{solid}, \left(8,0\right), \left(-2,5\right)\right\}\)
Answer 2 . 1.4.8.33.
Answer 1 .
\(\left\{\text{line}, \text{solid}, \left(3,0\right), \left(0,-2\right)\right\}\)
Answer 2 .
\(\text{infinitely many solutions}\)
1.4.8.35.
Answer 1 .
\(\left\{\text{line}, \text{solid}, \left(0,1\right), \left(7,2\right)\right\}, \left\{\text{line}, \text{solid}, \left(5,-1\right), \left(7,2\right)\right\}\)
Answer 2 .
\(\text{infinitely many solutions}\)
1.4.8.37.
Answer 1 .
\(\left\{\text{line}, \text{solid}, \left(0,-2\right), \left(7,-12\right)\right\}\)
Answer 2 .
1.4.8.39. 1.4.8.41. 1.4.8.43. 1.4.8.45. 1.4.8.47. 1.4.8.49. 1.4.8.51. Answer .
\(\text{infinitely many solutions}\)
1.4.8.53. 1.4.8.55. 1.4.8.57. 1.4.8.59. Answer .
\(B = \left({\frac{8}{7}}\right), n = \left(-{\frac{33}{7}}\right)\)
1.4.8.61. Answer .
\(\text{infinitely many solutions}\)
1.4.8.63. Answer .
\(Z = \left(-{\frac{51}{5}}\right), T = \left({\frac{17}{5}}\right)\)
1.4.8.65. Answer .
\(k = -7, j = \left({\frac{3}{2}}\right)\)
1.4.8.67. Answer .
\(v = \left({\frac{1}{4}}\right), A = \left({\frac{13}{2}}\right)\)
1.4.8.69. Answer .
\(H = \left(-{\frac{61}{9}}\right), Q = -7\)
1.4.8.71. Answer .
\(T = 1, f = \left({\frac{1}{2}}\right)\)
1.4.8.73. Answer .
\(e = 1.21739, w = -9.94783\)
1.4.8.75. Answer .
\(q = -1.59884, L = 0.308582\)
1.4.8.77. Answer .
\(B = -35, b = \left(-{\frac{125}{12}}\right)\)
1.4.8.79. Answer .
\(M = \left({\frac{20}{7}}\right), r = \left({\frac{52}{21}}\right)\)
1.4.8.81. Answer .
\(Z = \left(-{\frac{385}{27}}\right), G = \left({\frac{125}{27}}\right)\)
1.4.8.83. 1.4.8.85. Answer .
\(v = \left({\frac{105}{38}}\right), m = \left({\frac{3}{19}}\right)\)
1.4.8.87.
Answer 1 .
\(H = \left(-{\frac{49}{5}}\right), B = \left({\frac{49}{15}}\right)\)
Answer 2 . 1.4.8.89. Answer .
\(\text{infinitely many solutions}\)
Applications 1.4.8.91. 1.4.8.91.a 1.4.8.91.b Answer .
\(\left\{\text{line}, \text{solid}, \left(0,30\right), \left(20,0\right)\right\}, \left\{\text{line}, \text{solid}, \left(0,24\right), \left(24,0\right)\right\}\)
1.4.8.91.c 1.4.8.93. 1.4.8.93.a
Answer 1 . Answer 2 .
\(d = 55+\left(-8\right)t\)
1.4.8.93.b Answer .
\(\left\{\text{line}, \text{solid}, \left(0,0\right), \left(1,3\right)\right\}, \left\{\text{line}, \text{solid}, \left(0,55\right), \left(1,47\right)\right\}\)
1.4.8.93.c 1.4.8.95.
Answer 1 . Answer 2 . Answer 3 . Answer 4 . 1.4.8.97.
Answer 1 . Answer 2 . Answer 3 . Answer 4 . 1.4.8.99.
Answer 1 . Answer 2 . Answer 3 . Answer 4 . 1.4.8.101.
Answer 1 . Answer 2 .
\(0.02x-0.035y = -145.5\)
Answer 3 . Answer 4 . 1.4.8.103.
Answer 1 . Answer 2 . Answer 3 . Answer 4 .
2 Quadratic 2.1 Factoring Out the Common Factor 2.1.7 Exercises
2.1.7.1. 2.1.7.3. 2.1.7.5. 2.1.7.7.
2.1.7.9. 2.1.7.11. Answer .
\(3t\hbox{ or }-\left(3t\right)\)
2.1.7.13. Answer .
\(5x^{3}\hbox{ or }-5x^{3}\)
2.1.7.15. Answer .
\(8y^{10}\hbox{ or }-8y^{10}\)
2.1.7.17. Answer .
\(2r^{9}\hbox{ or }-2r^{9}\)
2.1.7.19. Answer .
\(9x^{4}y^{2}\hbox{ or }-9x^{4}y^{2}\)
2.1.7.21. Answer .
\(7\mathopen{}\left(x+1\right)\)
2.1.7.23. Answer .
\(10\mathopen{}\left(y-1\right)\)
2.1.7.25. Answer .
\(-4\mathopen{}\left(r+1\right)\)
2.1.7.27. Answer .
\(7\mathopen{}\left(r+5\right)\)
2.1.7.29. Answer .
\(3\mathopen{}\left(9t^{2}+10\right)\)
2.1.7.31. Answer .
\(4\mathopen{}\left(10x^{2}+10x+3\right)\)
2.1.7.33. Answer .
\(3y^{2}\mathopen{}\left(6y^{2}+9y+10\right)\)
2.1.7.35. Answer .
\(6r^{3}\mathopen{}\left(10r^{2}+3r+6\right)\)
2.1.7.37. Answer .
\(8t\mathopen{}\left(6+t+2t^{2}\right)\)
2.1.7.39. 2.1.7.41. Answer .
\(4y\mathopen{}\left(x+1\right)\)
2.1.7.43. Answer .
\(6y^{3}\mathopen{}\left(x^{5}-2\right)\)
2.1.7.45. Answer .
\(7x^{3}y^{10}\mathopen{}\left(8x^{2}-4x+9\right)\)
2.1.7.47. Answer .
\(2x^{3}y^{10}z^{8}\mathopen{}\left(10x^{2}z^{2}+2xz+5\right)\)
2.2 Factoring Trinomials with Leading Coefficient One 2.2.6 Exercises
2.2.6.1. 2.2.6.3. 2.2.6.5. 2.2.6.7. 2.2.6.9.
2.2.6.11. Answer .
\(\left(r+8\right)\mathopen{}\left(r+6\right)\)
2.2.6.13. Answer .
\(\left(r+1\right)\mathopen{}\left(r+5\right)\)
2.2.6.15. Answer .
\(\left(t-4\right)\mathopen{}\left(t+2\right)\)
2.2.6.17. Answer .
\(\left(x-7\right)\mathopen{}\left(x+6\right)\)
2.2.6.19. Answer .
\(\left(y-10\right)\mathopen{}\left(y-7\right)\)
2.2.6.21. Answer .
\(\left(r-3\right)\mathopen{}\left(r-8\right)\)
2.2.6.23. Answer .
\(\left(t+6\right)\mathopen{}\left(t+9\right)\)
2.2.6.25. Answer .
\(\left(x+9\right)\mathopen{}\left(x+5\right)\)
2.2.6.27. Answer .
\(\left(y-2\right)\mathopen{}\left(y+8\right)\)
2.2.6.29. Answer .
\(\left(r-5\right)\mathopen{}\left(r+3\right)\)
2.2.6.31. Answer .
\(\left(r-8\right)\mathopen{}\left(r-10\right)\)
2.2.6.33. Answer .
\(\left(t-1\right)\mathopen{}\left(t-8\right)\)
2.2.6.35. 2.2.6.37. 2.2.6.39. Answer .
\(\left(r+1\right)\mathopen{}\left(r+1\right)\)
2.2.6.41. Answer .
\(\left(t+4\right)\mathopen{}\left(t+4\right)\)
2.2.6.43. Answer .
\(\left(x-8\right)\mathopen{}\left(x-8\right)\)
2.2.6.45. Answer .
\(\left(y-12\right)\mathopen{}\left(y-12\right)\)
2.2.6.47. Answer .
\(2\mathopen{}\left(y-2\right)\mathopen{}\left(y+6\right)\)
2.2.6.49. Answer .
\(2\mathopen{}\left(r-9\right)\mathopen{}\left(r+1\right)\)
2.2.6.51. Answer .
\(8\mathopen{}\left(t-1\right)\mathopen{}\left(t-2\right)\)
2.2.6.53. Answer .
\(5\mathopen{}\left(x-1\right)\mathopen{}\left(x-4\right)\)
2.2.6.55. Answer .
\(3y^{4}\mathopen{}\left(y+1\right)\mathopen{}\left(y+7\right)\)
2.2.6.57. Answer .
\(4r^{2}\mathopen{}\left(r+5\right)\mathopen{}\left(r+1\right)\)
2.2.6.59. Answer .
\(3t^{5}\mathopen{}\left(t-2\right)\mathopen{}\left(t+2\right)\)
2.2.6.61. Answer .
\(3x^{7}\mathopen{}\left(x-4\right)\mathopen{}\left(x+4\right)\)
2.2.6.63. Answer .
\(2y^{6}\mathopen{}\left(y-10\right)\mathopen{}\left(y-1\right)\)
2.2.6.65. Answer .
\(4y^{8}\mathopen{}\left(y-3\right)\mathopen{}\left(y-1\right)\)
2.2.6.67. Answer .
\(-\left(r+3\right)\mathopen{}\left(r-7\right)\)
2.2.6.69. Answer .
\(-\left(t+6\right)\mathopen{}\left(t-2\right)\)
2.2.6.71. Answer .
\(\left(x+8t\right)\mathopen{}\left(x+2t\right)\)
2.2.6.73. Answer .
\(\left(y-8x\right)\mathopen{}\left(y+x\right)\)
2.2.6.75. Answer .
\(\left(r-y\right)\mathopen{}\left(r-3y\right)\)
2.2.6.77. Answer .
\(\left(t+12x\right)\mathopen{}\left(t+12x\right)\)
2.2.6.79. Answer .
\(\left(x-5t\right)\mathopen{}\left(x-5t\right)\)
2.2.6.81. Answer .
\(2\mathopen{}\left(y+2\right)\mathopen{}\left(y+1\right)\)
2.2.6.83. Answer .
\(4y\mathopen{}\left(x+1\right)\mathopen{}\left(x+2\right)\)
2.2.6.85. Answer .
\(4b\mathopen{}\left(a-1\right)\mathopen{}\left(a+5\right)\)
2.2.6.87. Answer .
\(2y\mathopen{}\left(x-6\right)\mathopen{}\left(x-1\right)\)
2.2.6.89. Answer .
\(2xy\mathopen{}\left(x+2\right)\mathopen{}\left(x+7\right)\)
2.2.6.91. Answer .
\(x^{2}\mathopen{}\left(y+z\right)\mathopen{}\left(y-7z\right)\)
2.2.6.93. Answer .
\(\left(r+0.3\right)\mathopen{}\left(r+0.8\right)\)
2.2.6.95. Answer .
\(\left(tr+7\right)\mathopen{}\left(tr+2\right)\)
2.2.6.97. Answer .
\(\left(xt-5\right)\mathopen{}\left(xt+7\right)\)
2.2.6.99. Answer .
\(\left(yx-8\right)\mathopen{}\left(yx-2\right)\)
2.2.6.101. Answer .
\(2\mathopen{}\left(yx+2\right)\mathopen{}\left(yx+3\right)\)
2.2.6.103. Answer .
\(8\mathopen{}\left(rt-1\right)\mathopen{}\left(rt+1\right)\)
2.2.6.105. Answer .
\(3y\mathopen{}\left(xy-2\right)\mathopen{}\left(xy-4\right)\)
2.2.6.107. Answer .
\(\left(a+b\right)\mathopen{}\left(x+4\right)\mathopen{}\left(x+7\right)\)
2.3 Factoring Special Polynomials 2.3.5 Exercises
2.3.5.1. 2.3.5.3. 2.3.5.5.
2.3.5.7. 2.3.5.9. 2.3.5.11.
2.3.5.13. Answer .
\(\left(x+2\right)\mathopen{}\left(x-2\right)\)
2.3.5.15. Answer .
\(\left(6y+1\right)\mathopen{}\left(6y-1\right)\)
2.3.5.17. Answer .
\(\left(rt+7\right)\mathopen{}\left(rt-7\right)\)
2.3.5.19. Answer .
\(\left(8tx+3\right)\mathopen{}\left(8tx-3\right)\)
2.3.5.21. Answer .
\(\left(5+x\right)\mathopen{}\left(5-x\right)\)
2.3.5.23. Answer .
\(\left(11+8x\right)\mathopen{}\left(11-8x\right)\)
2.3.5.25. Answer .
\(\left(y^{2}+12\right)\mathopen{}\left(y^{2}-12\right)\)
2.3.5.27. Answer .
\(\left(4r^{2}+7\right)\mathopen{}\left(4r^{2}-7\right)\)
2.3.5.29. Answer .
\(\left(t^{7}+8\right)\mathopen{}\left(t^{7}-8\right)\)
2.3.5.31. Answer .
\(\left(3x^{2}+11y^{2}\right)\mathopen{}\left(3x^{2}-11y^{2}\right)\)
2.3.5.33. Answer .
\(\left(x^{3}+5y^{6}\right)\mathopen{}\left(x^{3}-5y^{6}\right)\)
2.3.5.35. Answer .
\(\left(r+6\right)\mathopen{}\left(r+6\right)\)
2.3.5.37. Answer .
\(\left(t-10\right)\mathopen{}\left(t-10\right)\)
2.3.5.39. Answer .
\(\left(2x+1\right)\mathopen{}\left(2x+1\right)\)
2.3.5.41. Answer .
\(\left(6x-1\right)\mathopen{}\left(6x-1\right)\)
2.3.5.43. Answer .
\(\left(8yr-1\right)\mathopen{}\left(8yr-1\right)\)
2.3.5.45. Answer .
\(\left(r+6t\right)\mathopen{}\left(r+6t\right)\)
2.3.5.47. Answer .
\(\left(t-4y\right)\mathopen{}\left(t-4y\right)\)
2.3.5.49. Answer .
\(\left(3x+11r\right)\mathopen{}\left(3x+11r\right)\)
2.3.5.51. Answer .
\(\left(3y-8t\right)\mathopen{}\left(3y-8t\right)\)
2.3.5.53. Answer .
\(\left(r-2\right)\mathopen{}\left(r+2\right)\mathopen{}\left(r^{2}+4\right)\)
2.3.5.55. Answer .
\(10\mathopen{}\left(t+3\right)\mathopen{}\left(t-3\right)\)
2.3.5.57. Answer .
\(5x\mathopen{}\left(x+1\right)\mathopen{}\left(x-1\right)\)
2.3.5.59. Answer .
\(5xy\mathopen{}\left(xy+3\right)\mathopen{}\left(xy-3\right)\)
2.3.5.61. Answer .
\(3\mathopen{}\left(6+y\right)\mathopen{}\left(6-y\right)\)
2.3.5.63. Answer .
\(2\mathopen{}\left(2r+1\right)\mathopen{}\left(2r+1\right)\)
2.3.5.65. Answer .
\(10\mathopen{}\left(2tx+1\right)\mathopen{}\left(2tx+1\right)\)
2.3.5.67. Answer .
\(5\mathopen{}\left(3x-1\right)\mathopen{}\left(3x-1\right)\)
2.3.5.69. Answer .
\(y^{5}\mathopen{}\left(9y+1\right)\mathopen{}\left(9y+1\right)\)
2.3.5.71. Answer .
\(r^{4}\mathopen{}\left(12r-1\right)\mathopen{}\left(12r-1\right)\)
2.3.5.73. Answer .
\(3t^{8}\mathopen{}\left(4t+1\right)\mathopen{}\left(4t+1\right)\)
2.3.5.75. Answer .
\(9t^{5}\mathopen{}\left(2t-1\right)\mathopen{}\left(2t-1\right)\)
2.3.5.77. Answer .
\(3\mathopen{}\left(x-3\right)\mathopen{}\left(x+3\right)\mathopen{}\left(x^{2}+9\right)\)
2.3.5.79. 2.3.5.81. Answer .
\(10r\mathopen{}\left(r^{2}+4\right)\)
2.3.5.83. Answer .
\(t\mathopen{}\left(0.8+t\right)\mathopen{}\left(0.8-t\right)\)
2.4 Solving Quadratic Equations by Factoring 2.4.5 Exercises
2.4.5.1. Answer .
\(5\mathopen{}\left(y-10\right)\)
2.4.5.3. Answer .
\(\left(r+7\right)\mathopen{}\left(r-6\right)\)
2.4.5.5. Answer .
\(\left(2t-7\right)\mathopen{}\left(t-3\right)\)
2.4.5.7. Answer .
\(7\mathopen{}\left(4t^{2}-3t+8\right)\)
2.4.5.9. Answer .
\(\left(8x^{2}+11\right)\mathopen{}\left(8x^{2}-11\right)\)
2.4.5.11. 2.4.5.13. Answer .
\(\left\{-3,\frac{5}{6}\right\}\)
2.4.5.15. 2.4.5.17. 2.4.5.19. 2.4.5.21. 2.4.5.23. 2.4.5.25. Answer .
\(\left\{-10,-1\right\}\)
2.4.5.27. 2.4.5.29. 2.4.5.31. 2.4.5.33. 2.4.5.35. Answer .
\(\left\{0,\frac{-1}{2}\right\}\)
2.4.5.37. 2.4.5.39. 2.4.5.41. Answer .
\(\left\{\frac{-1}{9}\right\}\)
2.4.5.43. Answer .
\(\left\{\frac{-10}{3},-4\right\}\)
2.4.5.45. 2.4.5.47. Answer .
\(\left\{\frac{-5}{3},\frac{5}{3}\right\}\)
2.4.5.49. Answer .
\(\left\{\frac{-9}{7},\frac{9}{7}\right\}\)
2.4.5.51. 2.4.5.53. Answer .
\(\left\{\frac{5}{2},-10\right\}\)
2.4.5.55. 2.4.5.57. 2.4.5.59. 2.4.5.61. Answer .
\(\left\{\frac{-7}{10}\right\}\)
2.4.5.63. Answer .
\(\left\{-2,-9,-7\right\}\)
2.4.5.65. Answer .
\(\left\{-5,5,0\right\}\)
2.4.5.67. Answer .
\(\left\{0,-3,1\right\}\)
2.4.5.69. 2.4.5.71. 2.4.5.73. 2.4.5.75. 2.4.5.77. Answer .
\((x-8)*(x--1)*(x-2/3)=0\)
2.5 Solving Quadratic Equations by Using a Square Root 2.5.4 Exercises
Skills Practice
2.5.4.1. 2.5.4.3. Answer .
\(\left\{-\frac{1}{6},\frac{1}{6}\right\}\)
2.5.4.5. Answer .
\(\left\{2\sqrt{3},-2\sqrt{3}\right\}\)
2.5.4.7. Answer .
\(\left\{\sqrt{59},-\sqrt{59}\right\}\)
2.5.4.9. 2.5.4.11. Answer .
\(\left\{-\frac{2}{7},\frac{2}{7}\right\}\)
2.5.4.13. Answer .
\(\left\{-\frac{7}{4},\frac{7}{4}\right\}\)
2.5.4.15. Answer .
\(\left\{\frac{\sqrt{215}}{5},\frac{-\sqrt{215}}{5}\right\}\)
2.5.4.17. Answer .
\(\left\{\frac{-\sqrt{14}}{2},\frac{\sqrt{14}}{2}\right\}\)
2.5.4.19. Answer .
\(\text{no real solutions}\)
2.5.4.21. 2.5.4.23. Answer .
\(\left\{10,-14\right\}\)
2.5.4.25. Answer .
\(\left\{\frac{-15}{4},\frac{-5}{4}\right\}\)
2.5.4.27. Answer .
\(\left\{-8,-10\right\}\)
2.5.4.29. Answer .
\(\left\{-7+\sqrt{13},-7-\sqrt{13}\right\}\)
2.5.4.31. Answer .
\(\left\{-3+2\sqrt{7},-3-2\sqrt{7}\right\}\)
2.5.4.33. Answer .
\(\left\{9+3\sqrt{5},9-3\sqrt{5}\right\}\)
2.5.4.35. Answer .
\(t = \sqrt{\frac{H-h}{16}}\)
2.5.4.37. Answer .
\(r = \sqrt{x^{2}+y^{2}}\)
2.5.4.39. Answer .
\(x = \sqrt{2\mathopen{}\left(y-k\right)}+h\)
Applications
2.5.4.41. 2.5.4.43. 2.5.4.45. 2.5.4.47. 2.5.4.49.
2.5.4.51. 2.5.4.53. 2.5.4.55. 2.5.4.57. 2.5.4.59. Challenge 2.5.4.61.
2.6 The Quadratic Formula 2.6.5 Exercises
Review and Warmup
2.6.5.1. 2.6.5.1.a 2.6.5.1.b 2.6.5.1.c 2.6.5.1.d 2.6.5.3. 2.6.5.3.a 2.6.5.3.b 2.6.5.3.c 2.6.5.3.d Skills Practice
2.6.5.13. 2.6.5.15. Answer .
\(\left\{-9,\frac{2}{3}\right\}\)
2.6.5.17. Answer .
\(\left\{\frac{6}{5},\frac{3}{2}\right\}\)
2.6.5.19. Answer .
\(\text{no real solutions}\)
2.6.5.21. Answer .
\(\left\{-5,\frac{-5}{2}\right\}\)
2.6.5.23. 2.6.5.25. Answer .
\(\text{no real solutions}\)
2.6.5.27. Answer .
\(\left\{\frac{-3}{2},\frac{8}{9}\right\}\)
2.6.5.29. Answer .
\(\left\{-2-\sqrt{2},-2+\sqrt{2}\right\}\)
2.6.5.31. Answer .
\(\left\{\frac{-1}{2}\right\}\)
2.6.5.33. Answer .
\(\left\{-\frac{9}{2}-\frac{1}{2}\sqrt{73},-\frac{9}{2}+\frac{1}{2}\sqrt{73}\right\}\)
2.6.5.35. Answer .
\(\left\{-\frac{-3}{16}-\frac{1}{16}\sqrt{201},-\frac{-3}{16}+\frac{1}{16}\sqrt{201}\right\}\)
2.6.5.37. Answer .
\(\left\{\frac{-3}{2}\right\}\)
2.6.5.39. Answer .
\(\left\{-\frac{-1}{4}-\frac{1}{144}\sqrt{57},-\frac{-1}{4}+\frac{1}{144}\sqrt{57}\right\}\)
2.6.5.41. Answer .
\(\left\{\frac{-1}{2}-\frac{1}{2}\sqrt{13},\frac{-1}{2}+\frac{1}{2}\sqrt{13}\right\}\)
2.6.5.43. 2.6.5.45. 2.6.5.47. Answer .
\(\left\{-9.3,-7\right\}\)
2.6.5.49. Answer .
\(\left\{-0.5,6.6\right\}\)
2.6.5.51. Answer .
\(\left\{-3.80624,0.936677\right\}\)
2.6.5.53. 2.6.5.55. 2.6.5.57. Answer .
\(\left\{-5,\frac{-5}{6}\right\}\)
2.6.5.59. Answer .
\(\left\{-3,\frac{-1}{8}\right\}\)
2.6.5.61. Answer .
\(\left\{-2,\frac{4}{3}\right\}\)
2.6.5.63. Answer .
\(\left\{\frac{-3}{2},\frac{8}{5}\right\}\)
2.6.5.65. Answer .
\(\text{no real solutions}\)
2.6.5.67. Answer .
\(\left\{\frac{-3}{2},3\right\}\)
2.6.5.69. Answer .
\(\left\{\frac{1}{5},\frac{4}{3}\right\}\)
2.6.5.71. Answer .
\(\left\{-4-\sqrt{10},-4+\sqrt{10}\right\}\)
2.6.5.73. Answer .
\(\text{no real solutions}\)
2.6.5.75. Answer .
\(\left\{-9,\frac{-5}{4}\right\}\)
2.6.5.77. Answer .
\(\left\{\frac{-1}{6},1\right\}\)
2.6.5.79. Answer .
\(\left\{-\frac{-15}{14}-\frac{-1}{14}\sqrt{57},-\frac{-15}{14}+\frac{-1}{14}\sqrt{57}\right\}\)
2.6.5.81. Answer .
\(\left\{-3-\sqrt{7},-3+\sqrt{7}\right\}\)
2.6.5.83. Answer .
\(\left\{-\frac{3}{14}-\frac{-1}{14}\sqrt{93},-\frac{3}{14}+\frac{-1}{14}\sqrt{93}\right\}\)
2.6.5.85. 2.6.5.87. 2.6.5.89. Answer .
\(\text{no real solutions}\)
2.6.5.91. Answer .
\(\text{no real solutions}\)
2.6.5.93. 2.6.5.95. Answer .
\(\left\{\frac{-1}{2},7\right\}\)
2.6.5.97. Answer .
\(\left\{\frac{1}{5}\right\}\)
Applications 2.6.5.99. 2.6.5.101. 2.6.5.103. 2.6.5.105. 2.6.5.107. 2.6.5.109. Challenge 2.6.5.111. Answer .
\(\frac{-n-\sqrt{n^{2}-4mp}}{2m}, \frac{-n+\sqrt{n^{2}-4mp}}{2m}\)
2.7 Key Features of Quadratic Graphs 2.7.6 Exercises
2.7.6.1.
Answer 1 . Answer 2 . Answer 3 . Answer 4 . Answer 5 . Answer 6 . Answer 7 . Answer 8 . Answer 9 . Answer 10 . 2.7.6.3.
Answer 1 . Answer 2 . Answer 3 . Answer 4 . Answer 5 . Answer 6 . Answer 7 . Answer 8 . Answer 9 . Answer 10 .
2.7.6.9. 2.7.6.11. 2.7.6.13. 2.7.6.15. 2.7.6.17. 2.7.6.19.
Answer 1 . Answer 2 .
\(\left({\frac{1}{2}},{\frac{11}{4}}\right)\)
2.7.6.21. 2.7.6.23. 2.7.6.25. 2.7.6.27.
Answer 1 . Answer 2 .
\(\left(-\left(-4\right),-3\right)\)
2.7.6.45. 2.7.6.47. 2.7.6.49.
Answer 1 . Answer 2 . Answer 3 .
\(21012.5\ {\rm ft^{2}}\)
2.7.6.51.
Answer 1 . Answer 2 . Answer 3 .
\(23112.5\ {\rm ft^{2}}\)
2.7.6.53.
Answer 1 . Answer 2 . Answer 3 . 2.7.6.55.
Answer 1 . Answer 2 . Answer 3 . 2.7.6.57.
Answer 1 . Answer 2 . Answer 3 . Answer 4 .
2.8 Graphing Quadratic Expressions 2.8.6 Exercises
2.8.6.1. 2.8.6.3. 2.8.6.5. 2.8.6.7. Answer .
\(\left\{\frac{\sqrt{118}}{2},\frac{-\sqrt{118}}{2}\right\}\)
2.8.6.9. Answer .
\(\left\{\frac{5-3\sqrt{5}}{4},\frac{5+3\sqrt{5}}{4}\right\}\)
2.8.6.11. Answer .
\(\text{no real solutions}\)
2.8.6.13.
Answer 1 . Answer 2 .
\(\left(5,0\right), \left(3,0\right)\)
2.8.6.15.
Answer 1 . Answer 2 .
\(\left(3,0\right), \left(-3,0\right)\)
2.8.6.17.
Answer 1 . Answer 2 .
\(\left(0,0\right), \left(5,0\right)\)
2.8.6.19. 2.8.6.21. 2.8.6.23. 2.8.6.25.
Answer 1 . Answer 2 .
\(\left(-0.876894,0\right), \left(-9.12311,0\right)\)
2.8.6.27.
Answer 1 . Answer 2 .
\(\left(9.90833,0\right), \left(-0.908327,0\right)\)
2.8.6.29. 2.8.6.31.
Answer 1 . Answer 2 .
\(\left(-0.75,0\right), \left(-3,0\right)\)
2.8.6.53.
Answer 1 . Answer 2 . Answer 3 . Answer 4 . Answer 5 . Answer 6 . 2.8.6.55.
Answer 1 . Answer 2 . Answer 3 . Answer 4 . Answer 5 . Answer 6 . 2.8.6.57. 2.8.6.59. 2.8.6.61. 2.8.6.63. 2.8.6.65.
Answer 1 .
\(\left(\frac{-n-\sqrt{n^{2}-4p}}{2},0\right), \left(\frac{-n+\sqrt{n^{2}-4p}}{2},0\right)\)
Answer 2 . Answer 3 .
\(\left(\frac{-n}{2},\frac{4p-n^{2}}{4}\right)\)
2.9 Quadratic Graphs and Vertex Form 2.9.6 Exercises
2.9.6.1. 2.9.6.3. 2.9.6.5. Answer .
\(\left(t+1\right)\mathopen{}\left(t+5\right)\)
2.9.6.7. Answer .
\(2\mathopen{}\left(x+1\right)\mathopen{}\left(x+8\right)\)
2.9.6.9.
Answer 1 .
\(\left\{r \mid r\ge -1\right\}\)
Answer 2 .
\(\left[-1,\infty \right)\)
2.9.6.11.
Answer 1 .
\(\left\{z \mid z > 1\right\}\)
Answer 2 .
\(\left(1,\infty \right)\)
2.9.6.13. Answer .
\(-2;\,11;\,-1;\,5;\,0;\,1;\,1;\,-1;\,2;\,-1;\,3;\,1;\,4;\,5\)
2.9.6.15. Answer .
\(-2;\,-5;\,-1;\,-1;\,0;\,1;\,1;\,1;\,2;\,-1;\,3;\,-5;\,4;\,-11\)
2.9.6.17. Answer .
\(-2;\,4;\,-1;\,6;\,0;\,2;\,1;\,-8;\,2;\,-24;\,3;\,-46;\,4;\,-74\)
2.9.6.19. Answer .
\(-2;\,32;\,-1;\,40;\,0;\,44;\,1;\,44;\,2;\,40;\,3;\,32;\,4;\,20\)
2.9.6.29.
Answer 1 . Answer 2 . Answer 3 .
\(\left(0.267949,0\right), \left(3.73205,0\right)\)
Answer 4 .
\(\left(-\infty ,\infty \right)\)
Answer 5 .
\(\left[-3,\infty \right)\)
Answer 6 . Answer 7 .
\(\left\{-1.16228,5.16228\right\}\)
Answer 8 .
\(\left[-1.16228,5.16228\right]\)
2.9.6.31.
Answer 1 .
\(\left(-1.58333,-5.3125\right)\)
Answer 2 . Answer 3 .
\(\left(-3.30129,0\right), \left(0.134627,0\right)\)
Answer 4 .
\(\left(-\infty ,\infty \right)\)
Answer 5 .
\(\left[-5.3125,\infty \right)\)
Answer 6 . Answer 7 .
\(\left\{-3.85789,0.691224\right\}\)
Answer 8 .
\(\left(-\infty ,-3.85789\right]\cup \left[0.691224,\infty \right)\)
2.9.6.33.
Answer 1 .
\(\left(1.8,-3.22\right)\)
Answer 2 . Answer 3 .
\(\left(-0.737716,0\right), \left(4.33772,0\right)\)
Answer 4 .
\(\left(-\infty ,\infty \right)\)
Answer 5 .
\(\left[-3.22,\infty \right)\)
Answer 6 . Answer 7 .
\(\left\{-2.49418,6.09418\right\}\)
Answer 8 .
\(\left(-2.49418,6.09418\right)\)
2.9.6.35. 2.9.6.37. 2.9.6.39. 2.9.6.41. 2.9.6.43.
2.9.6.45. 2.9.6.47. 2.9.6.49. Answer .
\(\left(0.8,-0.5\right)\)
2.9.6.51. Answer .
\(2\mathopen{}\left(x-2\right)^{2}-2\)
2.9.6.53. Answer .
\(-2\mathopen{}\left(x-3\right)^{2}+5\)
2.9.6.55. Answer .
\(-3\mathopen{}\left(x-4\right)^{2}+3\)
2.9.6.57. Answer .
\(f\mathopen{}\left(x\right) = 6\mathopen{}\left(x-5\right)^{2}+6\)
2.9.6.59. Answer .
\(f\mathopen{}\left(x\right) = -8\mathopen{}\left(x--8\right)^{2}+3\)
2.9.6.61.
Answer 1 .
\(\left(-\infty ,\infty \right)\)
Answer 2 .
\(\left[-7,\infty \right)\)
2.9.6.63.
Answer 1 .
\(\left(-\infty ,\infty \right)\)
Answer 2 .
\(\left[1,\infty \right)\)
2.9.6.65.
Answer 1 .
\(\left(-\infty ,\infty \right)\)
Answer 2 .
\(\left[9,\infty \right)\)
2.9.6.67.
Answer 1 .
\(\left(-\infty ,\infty \right)\)
Answer 2 .
\(\left[\frac{-5}{6},\infty \right)\)
2.9.6.69.
Answer 1 .
\(\left(-\infty ,\infty \right)\)
Answer 2 .
\(\left[-2,\infty \right)\)
2.9.6.71. 2.9.6.73. 2.9.6.75.
Answer 1 .
\({\frac{3}{2}};\,\text{left}\)
Answer 2 .
\({\frac{5}{9}};\,\text{down}\)
2.9.6.77.
Answer 1 . Answer 2 .
\(\left(x-1\right)\mathopen{}\left(x-3\right)\)
2.9.6.79.
Answer 1 . Answer 2 .
\(\left(x+5\right)\mathopen{}\left(x-13\right)\)
2.9.6.81.
Answer 1 . Answer 2 .
\(\left(-2,0\right), \left(6,0\right)\)
2.9.6.83.
Answer 1 . Answer 2 .
\(\left(4,0\right), \left(3,0\right)\)
2.9.6.85.
Answer 1 . Answer 2 .
\(\left(0,0\right), \left(-9,0\right)\)
2.9.6.87. 2.9.6.89.
Answer 1 . Answer 2 .
\(\left(\frac{5}{4},0\right), \left(\frac{1}{9},0\right)\)
2.10 Completing the Square 2.10.6 Exercises
2.10.6.1. 2.10.6.3. Answer .
\(\left\{\frac{14}{9},\frac{-4}{9}\right\}\)
2.10.6.5. Answer .
\(\left\{6+\sqrt{2},6-\sqrt{2}\right\}\)
2.10.6.7. 2.10.6.9. 2.10.6.11. Answer .
\(\left\{\frac{-1+\sqrt{14}}{2},\frac{-1-\sqrt{14}}{2}\right\}\)
2.10.6.13. 2.10.6.15. 2.10.6.17. Answer .
\(\left\{5-\sqrt{34},5+\sqrt{34}\right\}\)
2.10.6.19. 2.10.6.21. 2.10.6.23. Answer .
\(\left\{-5-\sqrt{23},-5+\sqrt{23}\right\}\)
2.10.6.25. Answer .
\(\left\{\frac{-3}{2},\frac{-1}{6}\right\}\)
2.10.6.27. Answer .
\(\left\{1-\frac{1}{2}\sqrt{6},1+\frac{1}{2}\sqrt{6}\right\}\)
2.10.6.29.
Answer 1 .
\(H\mathopen{}\left(t\right) = \left(t-4\right)^{2}-13\)
Answer 2 . 2.10.6.31.
Answer 1 .
\(f\mathopen{}\left(x\right) = \left(x+\frac{9}{2}\right)^{2}-\frac{93}{4}\)
Answer 2 .
\(\left(\frac{-9}{2},\frac{-93}{4}\right)\)
2.10.6.33.
Answer 1 .
\(g\mathopen{}\left(y\right) = 7\mathopen{}\left(y+\frac{3}{2}\right)^{2}-\frac{71}{4}\)
Answer 2 .
\(\left(\frac{-3}{2},\frac{-71}{4}\right)\)
2.10.6.35.
Answer 1 .
\(\left(-\infty ,\infty \right)\)
Answer 2 .
\(\left[2,\infty \right)\)
2.10.6.37.
Answer 1 .
\(\left(-\infty ,\infty \right)\)
Answer 2 .
\(\left(-\infty ,-9\right]\)
2.10.6.39.
Answer 1 .
\(\left(-\infty ,\infty \right)\)
Answer 2 .
\(\left[-3,\infty \right)\)
2.10.6.41.
Answer 1 .
\(\left(-\infty ,\infty \right)\)
Answer 2 .
\(\left(-\infty ,4\right]\)
2.10.6.63. 2.10.6.65. 2.10.6.67. Answer .
\(\left(-\infty ,\frac{19}{9}\right]\)
2.10.6.69. Answer .
\(\left[-18,\infty \right)\)
2.10.6.71.
3 Rational Functions and Equations 3.1 Introduction to Rational Functions 3.1.3 Exercises
3.1.3.1.
Answer 1 . Answer 2 . Answer 3 . Answer 4 . 3.1.3.3.
3.1.3.7. Answer .
\(\left(-\infty ,5\right)\cup \left(5,\infty \right)\)
3.1.3.9. Answer .
\(\left(-\infty ,-9\right)\cup \left(-9,3\right)\cup \left(3,\infty \right)\)
3.1.3.11. Answer .
\(\left(-\infty ,-6\right)\cup \left(-6,0\right)\cup \left(0,\infty \right)\)
3.1.3.13. Answer .
\(\left(-\infty ,-8\right)\cup \left(-8,8\right)\cup \left(8,\infty \right)\)
3.1.3.15. Answer .
\(\left(-\infty ,0\right)\cup \left(0,\infty \right)\)
3.1.3.17. Answer .
\(\left(-\infty ,\infty \right)\)
3.1.3.19. Answer .
\(\left(-\infty ,-4\right)\cup \left(-4,\infty \right)\)
3.1.3.21. Answer .
\(\left(-\infty ,5\right)\cup \left(5,\infty \right)\)
3.1.3.23. Answer .
\(\left(-\infty ,-6\right)\cup \left(-6,2\right)\cup \left(2,\infty \right)\)
3.1.3.25.
Answer 1 .
\(\left(-\infty ,-1\right)\cup \left(-1,\infty \right)\)
Answer 2 .
\(\left(-\infty ,0\right)\cup \left(0,\infty \right)\)
3.1.3.27.
Answer 1 .
\(\left(-\infty ,1\right)\cup \left(1,\infty \right)\)
Answer 2 .
\(\left(-\infty ,2\right)\)
3.1.3.29. 3.1.3.31. 3.1.3.33. 3.1.3.35.
3.2 Multiplication and Division of Rational Expressions 3.2.5 Exercises
3.2.5.1. 3.2.5.3. 3.2.5.5. 3.2.5.7.
3.2.5.9. Answer .
\(\left(y+11\right)\mathopen{}\left(y-11\right)\)
3.2.5.11. Answer .
\(\left(r+2\right)\mathopen{}\left(r+7\right)\)
3.2.5.13. Answer .
\(\left(t-5\right)\mathopen{}\left(t-6\right)\)
3.2.5.15. Answer .
\(2\mathopen{}\left(x-1\right)\mathopen{}\left(x-4\right)\)
3.2.5.17. Answer .
\(5y^{4}\mathopen{}\left(y+1\right)\mathopen{}\left(y+2\right)\)
3.2.5.19. Answer .
\(\left(6r-1\right)\mathopen{}\left(6r-1\right)\)
3.2.5.21.
Answer 1 . Answer 2 . Answer 3 . Answer 4 .
3.2.5.25. Answer .
\(\frac{1}{x-3}, x\ne -4\)
3.2.5.27. Answer .
\(-\frac{9}{y+5}, y\ne 7\)
3.2.5.29. 3.2.5.31. 3.2.5.33. Answer .
\(-\frac{10}{x+6}, x\ne 0\)
3.2.5.35. Answer .
\(-\frac{y}{y-5}, y\ne 5\)
3.2.5.37. Answer .
\(-\frac{r}{r+1}, r\ne 1\)
3.2.5.39. Answer .
\(\frac{t}{t-2}, t\ne 3\)
3.2.5.41. Answer .
\(-\frac{5t+2}{2t+3}, t\ne -1\)
3.2.5.43. Answer .
\(-\frac{x-1}{x-2}, \left(-\infty ,\infty \right)\)
3.2.5.45. Answer .
\(-\frac{y+3}{y+4}, y\ne 4\)
3.2.5.47. Answer .
\(-\frac{3r-5}{6r+5}, r\ne -1\)
3.2.5.49. Answer .
\(-\frac{t^{4}}{t+6}, t\ne -1\)
3.2.5.51. Answer .
\(\frac{x+6}{x-3}, x\ne 2\hbox{ and }x\ne 0\)
3.2.5.53. Answer .
\(\frac{y^{2}+4y+16}{y+4}, y\ne 4\)
3.2.5.55. 3.2.5.57. 3.2.5.59. 3.2.5.61.
3.2.5.63. Answer .
\(\frac{1}{y-8}, y\ne -4\)
3.2.5.65. Answer .
\(\frac{r-2}{r-3}, r\ne -2\hbox{ and }r\ne 0\)
3.2.5.67. Answer .
\(\frac{t-5}{3t+2}, t\ne 5\hbox{ and }t\ne 0\)
3.2.5.69. Answer .
\(\frac{x}{x-3}, x\ne -0.2\hbox{ and }x\ne 0\)
3.2.5.73.
Answer 1 .
\(-\frac{r^{7}}{r+4}, \left(-\infty ,\infty \right)\)
Answer 2 .
\(-\frac{r}{r+4}, r\ne 0\)
3.2.5.75. Answer .
\(3\mathopen{}\left(t-4\right), t\ne -3\hbox{ and }t\ne -2\)
3.2.5.77. Answer .
\(\frac{t^{2}}{\left(t+2\right)\mathopen{}\left(t+6\right)}, t\ne 2\hbox{ and }t\ne 4\)
3.2.5.79. Answer .
\(-\frac{2\mathopen{}\left(x-5\right)}{3x\mathopen{}\left(x+1\right)}, x\ne -2\hbox{ and }x\ne 5\)
3.2.5.81. Answer .
\(-\frac{y-1}{9\mathopen{}\left(8y-7\right)}, y\ne -0.875\hbox{ and }y\ne 0.833333\hbox{ and }y\ne 0\)
3.2.5.83. Answer .
\(\frac{1}{4r\mathopen{}\left(r+4\right)}, \left(-\infty ,\infty \right)\)
3.2.5.85. 3.2.5.87. Answer .
\(\frac{1}{2}, x\ne -\left(-4\right)\)
3.2.5.89. Answer .
\(-\frac{1}{y-4}, y\ne 2.5\hbox{ and }y\ne -2.5\)
3.2.5.91. Answer .
\(r^{2}\mathopen{}\left(r+5\right), r\ne 0\hbox{ and }r\ne 6\hbox{ and }r\ne -5\)
3.2.5.93. Answer .
\(\frac{7p+10}{p-8}, p\ne 0\)
3.2.5.95. Answer .
\(\frac{k+10}{8\mathopen{}\left(k-10\right)}, k\ne 0\hbox{ and }k\ne -10\)
3.2.5.97. Answer .
\(\frac{x\mathopen{}\left(x-2\right)}{\left(x+4\right)\mathopen{}\left(x+3\right)}, x\ne 4\hbox{ and }x\ne 3\hbox{ and }x\ne 2\)
3.2.5.99. Answer .
\(\frac{4\mathopen{}\left(y+r\right)}{2y+r}\)
3.2.5.101. 3.2.5.103. Answer .
\(3\mathopen{}\left(t-4x\right)\)
3.2.5.105. 3.2.5.107. 3.2.5.109. Answer .
\(\frac{1}{r\mathopen{}\left(r-4x\right)}\)
3.2.5.111. Answer .
\(r\mathopen{}\left(ry-5\right)\)
3.2.5.113. Answer .
\(\frac{6r^{4}\mathopen{}\left(t+8r\right)}{t^{3}}\)
3.2.5.115. 3.2.5.117.
3.3 Addition and Subtraction of Rational Expressions 3.3.5 Exercises
3.3.5.1. 3.3.5.3. 3.3.5.5. 3.3.5.7.
3.3.5.9. Answer .
\(\left(r+6\right)\mathopen{}\left(r-6\right)\)
3.3.5.11. Answer .
\(\left(t+8\right)\mathopen{}\left(t+3\right)\)
3.3.5.13. Answer .
\(\left(x-2\right)\mathopen{}\left(x-6\right)\)
3.3.5.15. Answer .
\(2\mathopen{}\left(y-6\right)\mathopen{}\left(y-1\right)\)
3.3.5.17. 3.3.5.19. 3.3.5.21. Answer .
\(-\frac{1}{t+4}, t\ne 7\)
3.3.5.23. Answer .
\(-\frac{1}{x+3}, x\ne 6\)
3.3.5.25. Answer .
\(\frac{3y}{2}, \left(-\infty ,\infty \right)\)
3.3.5.27. Answer .
\(\frac{r-14}{\left(r+6\right)\mathopen{}\left(r+2\right)}, \left(-\infty ,\infty \right)\)
3.3.5.29. Answer .
\(\frac{4t-29}{\left(t+4\right)\mathopen{}\left(t-5\right)}, \left(-\infty ,\infty \right)\)
3.3.5.31. Answer .
\(\frac{1}{x+2}, x\ne 2\)
3.3.5.33. Answer .
\(\frac{1}{y+4}, y\ne 4\)
3.3.5.35. Answer .
\(-\frac{4}{r+2}, r\ne 2\)
3.3.5.37. Answer .
\(-\frac{4}{r-1}, r\ne -1\)
3.3.5.39. Answer .
\(\frac{t-5}{t}, t\ne 9\)
3.3.5.41. Answer .
\(\frac{x-4}{x}, x\ne 6\)
3.3.5.43. Answer .
\(\frac{3}{y-2}, y\ne -2\)
3.3.5.45. Answer .
\(-\frac{r}{r-1}, r\ne -1\)
3.3.5.47. Answer .
\(-\frac{t}{t-3}, t\ne 8\)
3.3.5.49. Answer .
\(\frac{4}{x-5}, x\ne 0\)
3.3.5.51. Answer .
\(\frac{-5y-27}{y+5}, \left(-\infty ,\infty \right)\)
3.3.5.53. Answer .
\(-\frac{6\mathopen{}\left(2r-9\right)}{\left(r+3\right)\mathopen{}\left(r-3\right)}, \left(-\infty ,\infty \right)\)
3.3.5.55. 3.3.5.57. 3.3.5.59. Answer .
\(\frac{-6x^{2}-20y^{3}}{15xy^{4}}\hbox{, }\frac{2\mathopen{}\left(-3x^{2}-10y^{3}\right)}{15xy^{4}}\hbox{, or }-\frac{2\mathopen{}\left(3x^{2}+10y^{3}\right)}{15xy^{4}}\)
3.3.5.61. 3.3.5.63.
3.4 Complex Fractions 3.4.3 Exercises
3.4.3.1. 3.4.3.3. 3.4.3.5. 3.4.3.7.
3.4.3.9. Answer .
\(\frac{4p+9}{p+8}, p\ne 0\)
3.4.3.11. Answer .
\(\frac{k+6}{10\mathopen{}\left(k-6\right)}, k\ne 0\hbox{ and }k\ne -6\)
3.4.3.13. Answer .
\(\frac{9x+1}{x\mathopen{}\left(x+9\right)}, \left(-\infty ,\infty \right)\)
3.4.3.15. Answer .
\(\frac{4y\mathopen{}\left(y-3\right)}{-y-3}, y\ne 0\hbox{ and }y\ne 3\)
3.4.3.17. Answer .
\(\frac{8\mathopen{}\left(6q-29\right)}{-q+13}, q\ne 5\)
3.4.3.19. Answer .
\(-\frac{k+9}{\left(k-4\right)\mathopen{}\left(k+3\right)}, k\ne 3\)
3.4.3.21. Answer .
\(\frac{3\mathopen{}\left(x+1\right)}{\left(4x+3\right)\mathopen{}\left(x-1\right)}, x\ne -1\)
3.4.3.23. Answer .
\(-\frac{2\mathopen{}\left(5b-7\right)\mathopen{}\left(b-2\right)}{2b-3}, b\ne 1\hbox{ and }b\ne 2\)
3.4.3.25. Answer .
\(\frac{5y^{2}+6y-20}{3y-14}, y\ne -2\hbox{ and }y\ne 2\)
3.4.3.27. Answer .
\(\frac{\left(r-1\right)\mathopen{}\left(r+100\right)}{\left(r+10\right)\mathopen{}\left(r-10\right)}, r\ne -100\)
3.4.3.29. 3.4.3.31. 3.4.3.33. 3.4.3.35. Answer .
\(\frac{3}{r\mathopen{}\left(x+15\right)}\hbox{ or }\frac{3}{rx+15r}\)
3.4.3.37.
3.5 Solving Rational Equations 3.5.5 Exercises
3.5.5.1. 3.5.5.3. 3.5.5.5. 3.5.5.7. 3.5.5.9. Answer .
\(\left\{-11,12\right\}\)
3.5.5.11.
3.5.5.15. 3.5.5.17. 3.5.5.19. 3.5.5.21. 3.5.5.23. 3.5.5.25. Answer .
\(\left\{\frac{11}{36}\right\}\)
3.5.5.27. 3.5.5.29. 3.5.5.31. Answer .
\({\rm no\ real\ solutions}\)
3.5.5.33. Answer .
\(\text{no real solutions}\)
3.5.5.35. Answer .
\(\left\{\frac{-9}{4},\frac{3}{2}\right\}\)
3.5.5.37. Answer .
\(\left\{-\left(-2\right)\right\}\)
3.5.5.39. 3.5.5.41.
3.5.5.43. 3.5.5.45. 3.5.5.47. 3.5.5.49.
3.5.5.51. 3.5.5.53. 3.5.5.55. 3.5.5.57.
3.5.5.59. Answer .
\(\left\{-2.38333,0.457872\right\}\)
3.5.5.61. Answer .
\(\left\{-1.53677,1.22187\right\}\)
3.5.5.63. Answer .
\(\left\{-0.607409,0.828675,3.17873\right\}\)
3.5.5.65. 3.5.5.67. 3.5.5.69. 3.5.5.71. 3.5.5.73. 3.5.5.75. 3.5.5.77. Answer .
\(1.96337\ {\textstyle\frac{\rm\mathstrut mi}{\rm\mathstrut hr}}\)
3.5.5.79. Answer .
\(184.829\ {\textstyle\frac{\rm\mathstrut mi}{\rm\mathstrut hr}}\)
3.5.5.81.
4 Radical Expressions and Equations 4.1 Introduction to Exponent Rules 4.1.4 Exercises
Review and Warmup
4.1.4.1. 4.1.4.1.a 4.1.4.1.b 4.1.4.1.c 4.1.4.1.d 4.1.4.3. 4.1.4.3.a 4.1.4.3.b 4.1.4.3.c 4.1.4.3.d 4.1.4.5. 4.1.4.5.a 4.1.4.5.b 4.1.4.5.c 4.1.4.5.d Skills Practice
4.1.4.7. 4.1.4.9. 4.1.4.11. 4.1.4.13. 4.1.4.15. 4.1.4.17. 4.1.4.19. 4.1.4.21. 4.1.4.23. 4.1.4.25. 4.1.4.27. 4.1.4.29. 4.1.4.31. 4.1.4.33. 4.1.4.35.
4.1.4.37. 4.1.4.39. 4.1.4.41.
4.1.4.43. 4.1.4.43.a 4.1.4.43.b 4.1.4.45. 4.1.4.45.a 4.1.4.45.b 4.1.4.47. 4.1.4.47.a 4.1.4.47.b 4.1.4.49. 4.1.4.49.a 4.1.4.49.b 4.1.4.51. 4.1.4.51.a 4.1.4.51.b
4.1.4.53. 4.1.4.55. 4.1.4.57. 4.1.4.59. 4.1.4.61. Applications 4.1.4.65. Challenge 4.1.4.67. 4.1.4.67.a 4.1.4.67.b 4.1.4.67.c 4.1.4.69.
Answer 1 . Answer 2 . Answer 3 .
4.2 More Exponent Rules 4.2.7 Exercises
Review and Warmup
4.2.7.1. 4.2.7.1.a 4.2.7.1.b 4.2.7.1.c 4.2.7.1.d 4.2.7.3. 4.2.7.3.a 4.2.7.3.b 4.2.7.3.c 4.2.7.3.d Skills Practice
4.2.7.5. 4.2.7.7. 4.2.7.9. 4.2.7.11. 4.2.7.13. 4.2.7.15. 4.2.7.17. 4.2.7.19. 4.2.7.21. 4.2.7.23. 4.2.7.25. 4.2.7.27. 4.2.7.29. 4.2.7.31. 4.2.7.33. Answer .
\(\frac{x^{16}}{4y^{18}z^{20}}\)
4.2.7.35. Answer .
\(\frac{-27x^{12}}{512y^{6}}\)
4.2.7.37. 4.2.7.39. 4.2.7.41. 4.2.7.43. 4.2.7.45. 4.2.7.47. 4.2.7.49. 4.2.7.51. 4.2.7.53. 4.2.7.55. Answer .
\(\frac{1}{x^{18}r^{2}}\)
4.2.7.57. 4.2.7.59. 4.2.7.61. 4.2.7.63. 4.2.7.65. 4.2.7.67. 4.2.7.69. 4.2.7.71. 4.2.7.73. 4.2.7.75. 4.2.7.77. 4.2.7.79. 4.2.7.81. 4.2.7.83. 4.2.7.85. 4.2.7.87. 4.2.7.89. 4.2.7.91. 4.2.7.93. 4.2.7.95. 4.2.7.97. 4.2.7.99. 4.2.7.101. 4.2.7.103. 4.2.7.105. 4.2.7.107. 4.2.7.109. 4.2.7.111. Answer .
\(\frac{1}{r^{28}y^{20}}\)
4.2.7.113. Answer .
\(\frac{t^{52}}{x^{20}}\)
4.2.7.115. 4.2.7.117. Answer .
\(\frac{r^{22}}{y^{28}}\)
4.2.7.119. Answer .
\(\frac{t^{18}}{r^{26}}\)
4.2.7.121. Answer .
\(\frac{5y^{2}}{27x^{24}z^{5}}\)
4.2.7.123. Answer .
\(\frac{1}{x^{30}y^{27}z^{36}}\)
Challenge 4.2.7.125. 4.2.7.127.
Answer 1 . Answer 2 . Answer 3 . Answer 4 .
4.3 Square and \(n\) th Root Properties 4.3.10 Exercises
4.3.10.3. 4.3.10.3.a 4.3.10.3.b 4.3.10.3.c 4.3.10.5. 4.3.10.5.a 4.3.10.5.b Answer .
\(\text{not a real number}\)
4.3.10.7. 4.3.10.7.a 4.3.10.7.b 4.3.10.7.c 4.3.10.9. 4.3.10.9.a 4.3.10.9.b 4.3.10.9.c 4.3.10.11. 4.3.10.11.a 4.3.10.11.b 4.3.10.11.c
4.3.10.15. 4.3.10.17. 4.3.10.19. Answer .
\(\text{not a real number}\)
4.3.10.21. Answer .
\(\text{not a real number}\hbox{ or }0.545455i\)
4.3.10.23. 4.3.10.25. 4.3.10.25.a 4.3.10.25.b
4.3.10.27. 4.3.10.29. 4.3.10.31. 4.3.10.33. 4.3.10.35. 4.3.10.37. 4.3.10.39. 4.3.10.41. 4.3.10.43. 4.3.10.45. 4.3.10.47. 4.3.10.49. 4.3.10.51. 4.3.10.53. 4.3.10.55. 4.3.10.57. 4.3.10.59. 4.3.10.61. Answer .
\(\sqrt{133}+\sqrt{119}\)
4.3.10.63. 4.3.10.65. 4.3.10.67. 4.3.10.69. 4.3.10.71. 4.3.10.73. 4.3.10.75. 4.3.10.77. 4.3.10.79.
4.3.10.81. 4.3.10.83. 4.3.10.85. 4.3.10.87. Answer .
\(\text{not a real number}\)
4.3.10.89. Answer .
\(\text{not a real number}\)
4.3.10.91. 4.3.10.93. 4.3.10.95. 4.3.10.97. 4.3.10.99. Answer .
\(\frac{1}{4}\sqrt[3]{11}\)
4.3.10.101. Answer .
\(\frac{2}{5}\sqrt[3]{11}\)
4.3.10.103. Answer .
\(\frac{2}{5}\sqrt[3]{9}\)
4.3.10.105. Answer .
\(\sqrt{37}\hbox{ or }-\sqrt{37}\)
4.3.10.107. 4.3.10.109. 4.3.10.111. 4.3.10.113. Answer .
\(\sqrt[4]{33}\hbox{ or }-\sqrt[4]{33}\)
4.3.10.115. 4.3.10.117.
4.4 Rationalizing the Denominator 4.4.4 Exercises
4.4.4.1. 4.4.4.3. 4.4.4.5. 4.4.4.7. 4.4.4.9. 4.4.4.11. 4.4.4.13. Answer .
\(\frac{3\sqrt{10}}{70}\)
4.4.4.15. 4.4.4.17. 4.4.4.19. 4.4.4.21. Answer .
\(\frac{\sqrt{195}}{15}\)
4.4.4.23. Answer .
\(\frac{6\sqrt{39}}{13}\)
4.4.4.25. 4.4.4.27. 4.4.4.29. Answer .
\(\frac{49-7\sqrt{5}}{44}\)
4.4.4.31. Answer .
\(\frac{30-5\sqrt{23}}{13}\)
4.4.4.33. Answer .
\(\frac{45-5\sqrt{7}-9\sqrt{5}+\sqrt{35}}{-74}\)
4.4.4.35. Answer .
\(\frac{32-8\sqrt{7}-4\sqrt{2}+\sqrt{14}}{-9}\)
4.5 Radical Expressions and Rational Exponents 4.5.4 Exercises
Review and Warmup
4.5.4.1. 4.5.4.3. 4.5.4.5. 4.5.4.7. 4.5.4.9. 4.5.4.11. 4.5.4.13. Skills Practice
4.5.4.15. 4.5.4.17. 4.5.4.19. 4.5.4.21. 4.5.4.23. 4.5.4.25. 4.5.4.27. 4.5.4.27.a 4.5.4.27.b 4.5.4.27.c 4.5.4.29. 4.5.4.29.a 4.5.4.29.b 4.5.4.29.c 4.5.4.31. 4.5.4.31.a 4.5.4.31.b 4.5.4.31.c 4.5.4.33. 4.5.4.33.a 4.5.4.33.b 4.5.4.33.c
4.5.4.39. 4.5.4.41. Answer .
\(\left(6c+7\right)^{\frac{1}{4}}\)
4.5.4.43. 4.5.4.45.
4.5.4.47. 4.5.4.49. 4.5.4.51. 4.5.4.53. 4.5.4.55. Answer .
\(\frac{1}{\sqrt[7]{m^{4}}}\)
4.5.4.57.
4.5.4.59. 4.5.4.61. 4.5.4.63. 4.5.4.65. 4.5.4.67. 4.5.4.69. 4.5.4.71. Applications
4.6 Solving Radical Equations 4.6.4 Exercises
Skills Practice
4.6.4.1. 4.6.4.3. 4.6.4.5. 4.6.4.7. Answer .
\(\text{no real solutions}\)
4.6.4.9. 4.6.4.11. 4.6.4.13. Answer .
\(\text{no real solutions}\)
4.6.4.15. Answer .
\(\left\{\frac{64}{9}\right\}\)
4.6.4.17. 4.6.4.19. Answer .
\(\left\{\frac{59}{5}\right\}\)
4.6.4.21. 4.6.4.23. Answer .
\(L = \frac{8T^{2}}{\pi ^{2}}\)
Applications Challenge 4.6.4.27. Answer .
\(\left\{\frac{1}{36}\right\}\)
