Answer:
x ∈ (-∞, 45/49] ∪ [53/49, +∞)
Explanation:
To find the solution set for the inequality 4 - 7 |7x - 7| ≤ 8, we can break it down into two cases based on the absolute value.
Case 1: When (7x - 7) ≥ 0
In this case, we can remove the absolute value and rewrite the inequality as 4 - 7(7x - 7) ≤ 8. Simplifying further, we get 4 - 49x + 49 ≤ 8, which becomes -49x + 53 ≤ 8. Rearranging the terms, we have -49x ≤ -45, and dividing by -49 (remembering to flip the inequality sign since we're dividing by a negative number), we get x ≥ 45/49.
Case 2: When (7x - 7) < 0
Here, we need to flip the inequality when removing the absolute value, resulting in 4 + 7(7x - 7) ≤ 8. Expanding and simplifying, we have 4 + 49x - 49 ≤ 8, which becomes 49x - 45 ≤ 8. Solving for x, we get x ≤ 53/49.
Combining the solutions from both cases, we have the solution set in interval notation:
x ∈ (-∞, 45/49] ∪ [53/49, +∞)
This means that x can take any value less than or equal to 45/49 or any value greater than or equal to 53/49, including both of these endpoints.