Section Exercises
1. When solving an inequality, explain what happened from Step 1 to Step 2:[latex]\begin{array}{ll}\text{Step 1}\hfill & -2x>6\hfill \\ \text{Step 2}\hfill & x<-3\hfill \end{array}[/latex]
2. When solving an inequality, we arrive at
[latex]\begin{array}{l}x+2< x+3\hfill \\ 2< 3\hfill \end{array}[/latex]
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
[latex]\begin{array}{l}x+2>x+3\hfill \\ 2>3\hfill \end{array}[/latex]
Explain what our solution set is.
5. Describe how to graph [latex]y=|x - 3|[/latex]
For the following exercises, solve the inequality. Write your final answer in interval notation.
6. [latex]4x - 7\le 9[/latex]
7. [latex]3x+2\ge 7x - 1[/latex]
8. [latex]-2x+3>x - 5[/latex]
9. [latex]4\left(x+3\right)\ge 2x - 1[/latex]
10. [latex]-\frac{1}{2}x\le \frac{-5}{4}+\frac{2}{5}x[/latex]
11. [latex]-5\left(x - 1\right)+3>3x - 4-4x[/latex]
12. [latex]-3\left(2x+1\right)>-2\left(x+4\right)[/latex]
13. [latex]\frac{x+3}{8}-\frac{x+5}{5}\ge \frac{3}{10}[/latex]
14. [latex]\frac{x - 1}{3}+\frac{x+2}{5}\le \frac{3}{5}[/latex]
For the following exercises, solve the inequality involving absolute value. Write your final answer in interval notation.
15. [latex]|x+9|\ge -6[/latex]
16. [latex]|2x+3|<7[/latex]
17. [latex]|3x - 1|>11[/latex]
18. [latex]|2x+1|+1\le 6[/latex]
19. [latex]|x - 2|+4\ge 10[/latex]
20. [latex]|-2x+7|\le 13[/latex]
21. [latex]|x - 7|<-4[/latex]
22. [latex]|x - 20|>-1[/latex]
23. [latex]|\frac{x - 3}{4}|<2[/latex]
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. [latex]-4<3x+2\le 18[/latex]
29. [latex]3x+1>2x - 5>x - 7[/latex]
30. [latex]3y<5 - 2y<7+y[/latex]
31. [latex]2x - 5<-11\text{ or }5x+1\ge 6[/latex]
32. [latex]x+7<x+2[/latex]
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. [latex]|x - 1|>2[/latex]
34. [latex]|x+3|\ge 5[/latex]
35. [latex]|x+7|\le 4[/latex]
36. [latex]|x - 2|<7[/latex]
37. [latex]|x - 2|<0[/latex]
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. [latex]x+3<3x - 4[/latex]
39. [latex]x - 2>2x+1[/latex]
40. [latex]x+1>x+4[/latex]
41. [latex]\frac{1}{2}x+1>\frac{1}{2}x - 5[/latex]
42. [latex]4x+1<\frac{1}{2}x+3[/latex]
For the following exercises, write the set in interval notation.
43. [latex]\{x|-1<x<3\}[/latex]
44. [latex]\{x|x\ge 7\}[/latex]
45. [latex]\{x|x<4\}[/latex]
46. [latex]\{x|x\text{ is all real numbers}\}[/latex]
For the following exercises, write the interval in set-builder notation.
47. [latex]\left(-\infty ,6\right)[/latex]
48. [latex]\left(4,+\infty \right)[/latex]
49. [latex]\left[-3,5\right)[/latex]
50. [latex]\left[-4,1\right]\cup \left[9,\infty \right)[/latex]
For the following exercises, write the set of numbers represented on the number line in interval notation.
51.
52.
53.
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. [latex]|x+2|-5< 2[/latex]
55. [latex]\frac{-1}{2}|x+2|< 4[/latex]
56. [latex]|4x+1|-3> 2[/latex]
57. [latex]|x - 4|< 3[/latex]
58. [latex]|x+2|\ge 5[/latex]
59. Solve [latex]|3x+1|=|2x+3|[/latex]
60. Solve [latex]{x}^{2}-x>12[/latex]
61. [latex]\frac{x - 5}{x+7}\le 0[/latex], [latex]x\ne -7[/latex]
62. [latex]p=-{x}^{2}+130x - 3000[/latex] 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 [latex]V=20T[/latex], 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 [latex]C=\$20+.30\left(x - 60\right)[/latex]. If you have to keep your bill lower than $50, what is the maximum calling minutes you can use?
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