1. The problem asks to solve the inequalities involving absolute values: (i) \(|x| < 2\) (ii) \(|x - 7| < 3\) (iii) \(|x + 1| < 1\) (iv) \(|2x - 1| < 3\) 2. Recall that for an inequality \(|A| < b\) where \(b > 0\), we have \(-b < A < b\). 3. Solve each inequality using this rule: (i) \(|x| < 2\) means \(-2 < x < 2\). (ii) \(|x - 7| < 3\) means \(-3 < x - 7 < 3\). Adding 7 to all parts: $$ -3 + 7 < x < 3 + 7 $$ $$ 4 < x < 10 $$ (iii) \(|x + 1| < 1\) means \(-1 < x + 1 < 1\). Subtracting 1 from all parts: $$ -1 - 1 < x < 1 - 1 $$ $$ -2 < x < 0 $$ (iv) \(|2x - 1| < 3\) means \(-3 < 2x - 1 < 3\). Add 1 to all parts: $$ -3 + 1 < 2x < 3 + 1 $$ $$ -2 < 2x < 4 $$ Now divide all parts by 2: $$ \frac{-2}{2} < x < \frac{4}{2} $$ $$ -1 < x < 2 $$ 4. Summary of the solution intervals: (i) \(x \in (-2, 2)\) (ii) \(x \in (4, 10)\) (iii) \(x \in (-2, 0)\) (iv) \(x \in (-1, 2)\)