Solve for \( x \). \( \frac{1}{2}\left(\frac{3}{5} x-\frac{1}{2}\right)>x+\frac{3}{8} \) \( x>\frac{25}{28} \) \( x<-\frac{25}{28} \) \( x<\frac{25}{28} \) \( x>-\frac{25}{28} \)

To solve the inequality \( \frac{1}{2}\left(\frac{3}{5} x-\frac{1}{2}\right)>x+\frac{3}{8} \), we first simplify the left side: 1. Multiply out the left-hand side: \[ \frac{1}{2} \left( \frac{3}{5} x - \frac{1}{2} \right) = \frac{3}{10} x - \frac{1}{4} \] Now the inequality becomes: \[ \frac{3}{10} x - \frac{1}{4} > x + \frac{3}{8} \] 2. Rearranging gives: \[ \frac{3}{10} x - x > \frac{1}{4} + \frac{3}{8} \] \[ -\frac{7}{10} x > \frac{5}{8} \] 3. Divide both sides by -1 (which flips the inequality): \[ \frac{7}{10} x < -\frac{5}{8} \] \[ x < -\frac{50}{56} \] \[ x < -\frac{25}{28} \] Thus, we conclude that: \( x < -\frac{25}{28} \). Now, for further options: - Since \( x<-\frac{25}{28} \) means we're looking at numbers less than this value, it's important to remember that as \( x \) increases towards \( -\frac{25}{28} \), it approaches but never touches it. - The boundary point \( x=-\frac{25}{28} \) itself is not included in the solution set, forming an open interval in the negative number system.