Answer: [19/2, -3/2]
Explanation:
To solve the inequality 3|2x - 8| ≥ 33, you need to consider two cases: one where the expression inside the absolute value is non-negative and one where it's negative.
Case 1: 2x - 8 is non-negative (greater than or equal to 0):
3(2x - 8) ≥ 33
Now, solve for x:
6x - 24 ≥ 33
Add 24 to both sides:
6x ≥ 57
Divide by 6:
x ≥ 57/6
Case 2: 2x - 8 is negative (less than 0):
3(-(2x - 8)) ≥ 33
Multiply -1 inside the absolute value:
-3(2x - 8) ≥ 33
Now, solve for x:
-6x + 24 ≥ 33
Subtract 24 from both sides:
-6x ≥ 9
Divide by -6 (and reverse the inequality sign because you're dividing by a negative number):
x ≤ -9/6
Now, simplify both fractions:
x ≥ 19/2 or x ≤ -3/2
In interval notation:
x ∈ [19/2, -3/2]
So, the solution in interval notation using reduced fractions is [19/2, -3/2].