Solve and write in interval notation: -3 ¡Ü (x − 4) / 2 < 4

Solve and write in interval notation: -7 < 4x − 3 ¡Ü 7

Solve and write in interval notation: -4x − 2 < 6 AND 3x ¡Ý -6

To solve and write in interval notation, we need to isolate the variable and find the range of values that satisfy the inequality.

1. -3 ≤ (x − 4) / 2 < 4

Step 1: Multiply both sides of the inequality by 2 to eliminate the fraction.
-6 ≤ x - 4 < 8

Step 2: Add 4 to all parts of the inequality.
-6 + 4 ≤ x < 8 + 4
-2 ≤ x < 12

Interval notation: [-2, 12)

2. -7 < 4x − 3 ≤ 7

Step 1: Add 3 to all parts of the inequality.
-7 + 3 < 4x ≤ 7 + 3
-4 < 4x ≤ 10

Step 2: Divide all parts of the inequality by 4.
-4/4 < 4x/4 ≤ 10/4
-1 < x ≤ 5/2

Interval notation: (-1, 5/2]

3. -4x - 2 < 6 AND 3x ≥ -6

Solving the first inequality:
Step 1: Add 2 to all parts of the inequality.
-4x - 2 + 2 < 6 + 2
-4x < 8

Step 2: Divide all parts of the inequality by -4. (Note: When dividing by a negative number, the direction of the inequality sign changes.)
-4x / -4 > 8 / -4
x > -2

Interval notation: (-2, ∞)

Solving the second inequality:
Step 1: Divide both sides of the inequality by 3.
3x / 3 ≥ -6 / 3
x ≥ -2

Interval notation: [-2, ∞)

Combine the solutions:
Since the two inequalities use different direction signs, we need to find the intersection of their solution sets.

Intersection: (-2, ∞)

Note: The interval notation represents an open interval using parentheses () when the endpoint is not included and a closed interval using brackets [] when the endpoint is included.