Simplifying Interval Notation and Solving Simple Inequality Problems

A. Simplify each of the following:

1.) (-∞, 4)∪(2, 5]

2.) (3, 5)∩(5, 7]

3.) (4, 7)∩(6, 9)

4.) {(4, 7]∩[7, 8)}∪(3, 6)

5.) (4, 7]∩{[7, 8)∪(3, 6)}

6.) (4, 7]∪[7, 8)∪(3, 6)		 7.) {(4, 7]∪[7, 8)}∩(3, 6)

8.) (4, 7)∪{[7, 8)∩(3, 6)}

9.) {(4, 7]∩(3, 6)}∪[7, 8)

10.) {(4, 7]∪(3, 6)}∩[7, 8)

11.) (4, 7]∩{(3, 6)∪[7, 8)}

12.) (4, 7]∪{(3, 6)∩[7, 8)}

B. Solve and write your answer in interval notation.
  1. 2 - x ≥ 0,   x ≠ -4
  2. x + 1 ≥ 0,   x ≠ 2, x ≠ -3
  3. x - 3 ≥ 0,   x2 + 4x - 5 ≠ 0
  4. 2x + 5 ≥ 0,   x2 + 5x + 6 ≠ 0
  1. x2 + 1 ≥ 0
  2. x2 + 4x - 5 ≥ 0
  3. x2 + 5x + 6 ≤ 0
  4. 2x2 + x - 6 > 0

Solutions:
A.
1.) (-∞, 5]
2.) Ø
3.) (6, 7)
4.) (3, 6)∪{7} 		5.) (4, 6)∪{7}
6.) (3, 8)
7.) (4, 6)
8.) (4, 7] 		9.) (4, 6)∪[7, 8)
10.) {7}
11.) (4, 6)∪{7}
12.) (4, 7]
B.
  1. (-∞, -4)∪(-4, 2]
  2. [-1, 2)∪(2, ∞)
  3. [3, ∞)
  4. [-5/2, -2)∪(-2, ∞)
  1. (-∞, ∞)
  2. (-∞, -5]∪[1, ∞)
  3. [-3, -2]
  4. (-∞, -2)∪(3/2, ∞)