The solution set for x satisfying theinequation ||x−4|−5|<1, is

To solve the inequality ||x−4|−5|<1, we need to consider two cases due to the absolute value.

Case 1: If |x-4|-5 is positive or zero, then we have |x-4|-5<1. Adding 5 to both sides gives |x-4|<6. This implies -6<x-4<6. Adding 4 to all parts of the inequality gives -2<x<10.

Case 2: If |x-4|-5 is negative, then we have -(|x-4|-5)<1. This simplifies to 5-|x-4|<1. Subtracting 5 from both sides gives -|x-4|<-4. Multiplying all parts of the inequality by -1 (and reversing the inequality signs) gives |x-4|>4. This implies x-4>4 or x-4<-4. Solving these gives x>8 or x<0.

Combining the solutions from both cases, the solution set for the inequality ||x−4|−5|<1 is (-2, 10) union (-infinity, 0) union (8, infinity). However, we need to check the endpoints to make sure they satisfy the original inequality. Checking x=-2, 0, 8, and 10 in the original inequality, we find that x=0 and x=8 do not satisfy the inequality. Therefore, the solution set is (-2, 10) union (-infinity, 0) union (8, infinity) excluding {0, 8}.