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+7x-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

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. [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?