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Study Guides > College Algebra

Section Exercises

1. When solving an inequality, explain what happened from Step 1 to Step 2:
Step 1−2x>6Step 2x<−3\begin{array}{ll}\text{Step 1}\hfill & -2x>6\hfill \\ \text{Step 2}\hfill & x<-3\hfill \end{array}
2. When solving an inequality, we arrive at
x+2<x+32<3\begin{array}{l}x+2< x+3\hfill \\ 2< 3\hfill \end{array}
Explain what our solution set is. 3. When writing our solution in interval notation, how do we represent all the real numbers? 4. When solving an inequality, we arrive at
x+2>x+32>3\begin{array}{l}x+2>x+3\hfill \\ 2>3\hfill \end{array}
Explain what our solution set is. 5. Describe how to graph y=∣x−3∣y=|x - 3| For the following exercises, solve the inequality. Write your final answer in interval notation. 6. 4x−7≤94x - 7\le 9 7. 3x+2≥7x−13x+2\ge 7x - 1 8. −2x+3>x−5-2x+3>x - 5 9. 4(x+3)≥2x−14\left(x+3\right)\ge 2x - 1 10. −12x≤−54+25x-\frac{1}{2}x\le \frac{-5}{4}+\frac{2}{5}x 11. −5(x−1)+3>3x−4−4x-5\left(x - 1\right)+3>3x - 4-4x 12. −3(2x+1)>−2(x+4)-3\left(2x+1\right)>-2\left(x+4\right) 13. x+38−x+55≥310\frac{x+3}{8}-\frac{x+5}{5}\ge \frac{3}{10} 14. x−13+x+25≤35\frac{x - 1}{3}+\frac{x+2}{5}\le \frac{3}{5} For the following exercises, solve the inequality involving absolute value. Write your final answer in interval notation. 15. ∣x+9∣≥−6|x+9|\ge -6 16. ∣2x+3∣<7|2x+3|<7 17. ∣3x−1∣>11|3x - 1|>11 18. ∣2x+1∣+1≤6|2x+1|+1\le 6 19. ∣x−2∣+4≥10|x - 2|+4\ge 10 20. ∣−2x+7∣≤13|-2x+7|\le 13 21. ∣x−7∣<−4|x - 7|<-4 22. ∣x−20∣>−1|x - 20|>-1 23. ∣x−34∣<2|\frac{x - 3}{4}|<2 For the following exercises, describe all the x-values within or including a distance of the given values. 24. Distance of 5 units from the number 7 25. Distance of 3 units from the number 9 26. Distance of 10 units from the number 4 27. Distance of 11 units from the number 1 For the following exercises, solve the compound inequality. Express your answer using inequality signs, and then write your answer using interval notation. 28. −4<3x+2≤18-4<3x+2\le 18 29. 3x+1>2x−5>x−73x+1>2x - 5>x - 7 30. 3y<5−2y<7+y3y<5 - 2y<7+y 31. 2x−5<−11 or 5x+1≥62x - 5<-11\text{ or }5x+1\ge 6 32. x+7<x+2x+7<x+2 For the following exercises, graph the function. Observe the points of intersection and shade the x-axis representing the solution set to the inequality. Show your graph and write your final answer in interval notation. 33. ∣x−1∣>2|x - 1|>2 34. ∣x+3∣≥5|x+3|\ge 5 35. ∣x+7∣≤4|x+7|\le 4 36. ∣x−2∣<7|x - 2|<7 37. ∣x−2∣<0|x - 2|<0 For the following exercises, graph both straight lines (left-hand side being y1 and right-hand side being y2) on the same axes. Find the point of intersection and solve the inequality by observing where it is true comparing the y-values of the lines. 38. x+3<3x−4x+3<3x - 4 39. x−2>2x+1x - 2>2x+1 40. x+1>x+4x+1>x+4 41. 12x+1>12x−5\frac{1}{2}x+1>\frac{1}{2}x - 5 42. 4x+1<12x+34x+1<\frac{1}{2}x+3 For the following exercises, write the set in interval notation. 43. {x∣−1<x<3}\{x|-1<x<3\} 44. {x∣x≥7}\{x|x\ge 7\} 45. {x∣x<4}\{x|x<4\} 46. {x∣x is all real numbers}\{x|x\text{ is all real numbers}\} For the following exercises, write the interval in set-builder notation. 47. (−∞,6)\left(-\infty ,6\right) 48. (4,+∞)\left(4,+\infty \right) 49. [−3,5)\left[-3,5\right) 50. [−4,1]∪[9,∞)\left[-4,1\right]\cup \left[9,\infty \right) For the following exercises, write the set of numbers represented on the number line in interval notation. 51. A number line with two tick marks labeled: -2 and 1 respectively. There is an open circle around the -2 and a dot on the 1 with an arrow connecting the two. 52. Number line with two tick marks labeled: -1 and 2 respectively. There is an open circle around the tick mark labeled -1 and a line that extends leftward from the circle. There is a dot around the tick mark labeled 2 and a line that extends rightward from the dot. 53. Number line with one tick mark labeled 4. There is a dot on the tick mark and an arrow extending leftward from the dot. For the following exercises, input the left-hand side of the inequality as a Y1 graph in your graphing utility. Enter y2 = the right-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, 1:abs(. Find the points of intersection, recall (2nd CALC 5:intersection, 1st curve, enter, 2nd curve, enter, guess, enter). Copy a sketch of the graph and shade the x-axis for your solution set to the inequality. Write final answers in interval notation. 54. ∣x+2∣−5<2|x+2|-5< 2 55. −12∣x+2∣<4\frac{-1}{2}|x+2|< 4 56. ∣4x+1∣−3>2|4x+1|-3> 2 57. ∣x−4∣<3|x - 4|< 3 58. ∣x+2∣≥5|x+2|\ge 5 59. Solve ∣3x+1∣=∣2x+3∣|3x+1|=|2x+3| 60. Solve x2−x>12{x}^{2}-x>12 61. x−5x+7≤0\frac{x - 5}{x+7}\le 0, x≠−7x\ne -7 62. p=−x2+130x−3000p=-{x}^{2}+130x - 3000 is a profit formula for a small business. Find the set of x-values that will keep this profit positive. 63. In chemistry the volume for a certain gas is given by V=20TV=20T, where V is measured in cc and T is temperature in ºC. If the temperature varies between 80ºC and 120ºC, find the set of volume values. 64. A basic cellular package costs $20/mo. for 60 min of calling, with an additional charge of $.30/min beyond that time. The cost formula would be C=$20+.30(x−60)C=\$20+.30\left(x - 60\right). If you have to keep your bill lower than $50, what is the maximum calling minutes you can use?

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