|(|2х-4|-5)|=9

To solve the equation \(|(|2x-4|-5)|=9\), we will break it down step-by-step. ### Step 1: Analyze the Outer Absolute Value The equation \(|A| = 9\) implies that \(A = 9\) or \(A = -9\). Here, \(A = |2x-4|-5\). ### Step 2: Set Up Two Cases So, we have two cases to consider: 1. \(|2x - 4| - 5 = 9\) 2. \(|2x - 4| - 5 = -9\) ### Case 1: \(|2x - 4| - 5 = 9\) Add 5 to both sides: \[ |2x - 4| = 14 \] This leads to two subcases: 1. \(2x - 4 = 14\) 2. \(2x - 4 = -14\) **Subcase 1.1:** \[ 2x - 4 = 14 \\ 2x = 18 \\ x = 9 \] **Subcase 1.2:** \[ 2x - 4 = -14 \\ 2x = -10 \\ x = -5 \] ### Case 2: \(|2x - 4| - 5 = -9\) Add 5 to both sides: \[ |2x - 4| = -4 \] Since the absolute value cannot be negative, there are no solutions from this case. ### Step 3: Final Solutions The solutions from Case 1 are \(x = 9\) and \(x = -5\). Thus, the solutions to the equation \(|(|2x-4|-5)|=9\) are: \[ \boxed{9 \text{ and } -5} \]