Practice Makes Perfect
Solve Compound Inequalities with “and”
In the following exercises, solve each inequality, graph the solution, and write the solution in interval notation.
1. \(x<3\) and \(x\geq 1\)
2. \(x\leq 4\) and \(x>−2\)
 Answer

3. \(x\geq −4\) and \(x\leq −1\)
4. \(x>−6\) and \(x<−3\)
 Answer

5. \(5x−2<8\) and \(6x+9\geq 3\)
6. \(4x−1<7\) and \(2x+8\geq 4\)
 Answer

7. \(4x+6\leq 2\) and \(2x+1\geq −5\)
8. \(4x−2\leq 4\) and \(7x−1>−8\)
 Answer

9. \(2x−11<5\) and \(3x−8>−5\)
10. \(7x−8<6\) and \(5x+7>−3\)
 Answer

11. \(4(2x−1)\leq 12\) and \(2(x+1)<4\)
12. \(5(3x−2)\leq 5\) and \(3(x+3)<3\)
 Answer

13. \(3(2x−3)>3\) and \(4(x+5)\geq 4\)
14. \(−3(x+4)<0\) and \(−1(3x−1)\leq 7\)
 Answer

15. \(\frac{1}{2}(3x−4)\leq 1\) and \(\frac{1}{3}(x+6)\leq 4\)
16. \(\frac{3}{4}(x−8)\leq 3\) and \(\frac{1}{5}(x−5)\leq 3\)
 Answer

17. \(5x−2\leq 3x+4\) and \(3x−4\geq 2x+1\)
18. \(\frac{3}{4}x−5\geq −2\) and \(−3(x+1)\geq 6\)
 Answer

19. \(\frac{2}{3}x−6\geq −4\) and \(−4(x+2)\geq 0\)
20. \(\frac{1}{2}(x−6)+2<−5\) and \(4−\frac{2}{3}x<6\)
 Answer

22. \(−3<2x−5\leq 1\)
 Answer

24. \(−1<3x+2<8\)
 Answer

26. \(−6\leq 4x−2<−2\)
 Answer

Solve Compound Inequalities with “or”
In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
27. \(x\leq −2\) or \(x>3\)
28. \(x\leq −4\) or \(x>−3\)
 Answer

29. \(x<2\) or \(x\geq 5\)
30. \(x<0\) or \(x\geq 4\)
 Answer

31. \(2+3x\leq 4\) or \(5−2x\leq −1\)
32. \(4−3x\leq −2\) or \(2x−1\leq −5\)
 Answer

33. \(2(3x−1)<4\) or \(3x−5>1\)
34. \(3(2x−3)<−5\) or
\(4x−1>3\)
 Answer

35. \(\frac{3}{4}x−2>4\) or \(4(2−x)>0\)
36. \(\frac{2}{3}x−3>5\) or \(3(5−x)>6\)
 Answer

37. \(3x−2>4\) or \(5x−3\leq 7\)
38. \(2(x+3)\geq 0\) or \(3(x+4)\leq 6\)
 Answer

39. \(\frac{1}{2}x−3\leq 4\) or \(\frac{1}{3}(x−6)\geq −2\)
40. \(\frac{3}{4}x+2\leq −1\) or \(\frac{1}{2}(x+8)\geq −3\)
 Answer

Mixed practice
In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
41. \(3x+7\leq 1\) and \(2x+3\geq −5\)
42. \(6(2x−1)>6\) and \(5(x+2)\geq 0\)
 Answer

43. \(4−7x\geq −3\) or \(5(x−3)+8>3\)
44. \(\frac{1}{2}x−5\leq 3\) or \(\frac{1}{4}(x−8)\geq −3\)
 Answer

46. \(\frac{1}{5}(x−5)+6<4\) and \(3−\frac{2}{3}x<5\)
 Answer

47. \(4x−2>6\) or \(3x−1\leq −2\)
48. \(6x−3\leq 1\) and \(5x−1>−6\)
 Answer

49. \(−2(3x−4)\leq 2\) and \(−4(x−1)<2\)
50. \(−5\leq 3x−2\leq 4\)
 Answer

Solve Applications with Compound Inequalities
In the following exercises, solve.
51. Penelope is playing a number game with her sister June. Penelope is thinking of a number and wants June to guess it. Five more than three times her number is between 2 and 32. Write a compound inequality that shows the range of numbers that Penelope might be thinking of.
52. Gregory is thinking of a number and he wants his sister Lauren to guess the number. His first clue is that six less than twice his number is between four and fortytwo. Write a compound inequality that shows the range of numbers that Gregory might be thinking of.
 Answer

\(5\leq n\leq 24\)
53. Andrew is creating a rectangular dog run in his back yard. The length of the dog run is 18 feet. The perimeter of the dog run must be at least 42 feet and no more than 72 feet. Use a compound inequality to find the range of values for the width of the dog run.
54. Elouise is creating a rectangular garden in her back yard. The length of the garden is 12 feet. The perimeter of the garden must be at least 36 feet and no more than 48 feet. Use a compound inequality to find the range of values for the width of the garden.
 Answer

\(6\leq w\leq 12\)