Final answer:
To solve the inequality 2|3x - 1| - 1 < 7, we isolate the absolute value expression and solve two separate inequalities. The solution set is -1 < x < 5/3.
Step-by-step explanation:
To find the solution set for the inequality 2|3x - 1| - 1 < 7, we need to isolate the absolute value expression and solve two separate inequalities. First, we remove the outermost absolute value by adding 1 to both sides:
2|3x - 1| - 1 + 1 < 7 + 1
This simplifies to 2|3x - 1| < 8. Now we have two cases to consider: when 3x - 1 is positive and when it is negative. First, let's assume 3x - 1 > 0:
2(3x - 1) < 8
Expanding and solving this inequality gives us 6x - 2 < 8, which simplifies to 6x < 10. Dividing both sides by 6, we find x < 10/6 or x < 5/3. Next, let's assume 3x - 1 < 0:
2(-(3x - 1)) < 8
Expanding and solving this inequality gives us -6x + 2 < 8, which simplifies to -6x < 6. Dividing both sides by -6 and reversing the inequality sign, we get x > -1. Therefore, the solution set for the original inequality is -1 < x < 5/3.