(b) Solve the inequality \( \left|3-\frac{1}{x}\right|<\frac{1}{2} \) and give your answer in interval notation.

To solve the inequality \( \left|3-\frac{1}{x}\right|<\frac{1}{2} \), we split it into two cases based on the definition of absolute value: 1. \( 3 - \frac{1}{x} < \frac{1}{2} \) 2. \( 3 - \frac{1}{x} > -\frac{1}{2} \) ### Case 1: \( 3 - \frac{1}{x} < \frac{1}{2} \) Rearranging it, we have: \[ -\frac{1}{x} < \frac{1}{2} - 3 \] \[ -\frac{1}{x} < -\frac{5}{2} \] Multiplying both sides by -1 (which flips the inequality): \[ \frac{1}{x} > \frac{5}{2} \] This gives: \[ x < \frac{2}{5} \] ### Case 2: \( 3 - \frac{1}{x} > -\frac{1}{2} \) Rearranging this inequality: \[ -\frac{1}{x} > -\frac{1}{2} - 3 \] \[ -\frac{1}{x} > -\frac{7}{2} \] Again, multiplying by -1 (flipping the inequality): \[ \frac{1}{x} < \frac{7}{2} \] This leads to: \[ x > \frac{2}{7} \] ### Combine the results Thus, we have two conditions: 1. \( x < \frac{2}{5} \) 2. \( x > \frac{2}{7} \) Since we are looking for solutions where both conditions hold, the solution in interval notation becomes: \[ \left(\frac{2}{7}, \frac{2}{5}\right) \] This is your final answer!