\( A_{2}=\{n \in \mathbb{N}: 3+2|4 n-3|>13\} \) each set, give TWO elements belonging

We start with the inequality \[ 3 + 2\lvert 4n - 3 \rvert > 13. \] **Step 1: Subtract 3 from both sides** \[ 2\lvert 4n - 3 \rvert > 10. \] **Step 2: Divide both sides by 2** \[ \lvert 4n - 3 \rvert > 5. \] **Step 3: Solve the absolute value inequality** Recall that \(\lvert x \rvert > 5\) means either \(x > 5\) or \(x < -5\). - For \(4n - 3 > 5\): \[ 4n > 8 \quad \Rightarrow \quad n > 2. \] - For \(4n - 3 < -5\): \[ 4n < -2 \quad \Rightarrow \quad n < -\frac{1}{2}. \] Since \(n \in \mathbb{N}\) (the natural numbers), the inequality \(n < -\frac{1}{2}\) gives no solutions. Thus, the solution is given by \[ n > 2. \] **Step 4: List two elements of the set** Since \(n\) must be a natural number greater than 2, the smallest two numbers are: \[ n = 3, \quad n = 4. \]