To determine which equation has the solution of \( x = -\frac{11}{4} \), we can substitute \( x = -\frac{11}{4} \) into each option and check if both sides are equal.
a. \( -\frac{1}{2}x = \frac{11}{9} \)
Substituting \( x = -\frac{11}{4} \): \[ -\frac{1}{2}\left(-\frac{11}{4}\right) = \frac{11}{8} \quad (\text{not } \frac{11}{9}) \]
b. \( -2x = \frac{2}{11} \)
Substituting \( x = -\frac{11}{4} \): \[ -2\left(-\frac{11}{4}\right) = \frac{22}{4} = \frac{11}{2} \quad (\text{not } \frac{2}{11}) \]
c. \( \frac{1}{2}x = -11 \)
Substituting \( x = -\frac{11}{4} \): \[ \frac{1}{2}\left(-\frac{11}{4}\right) = -\frac{11}{8} \quad (\text{not } -11) \]
d. \( -2x = \frac{11}{2} \)
Substituting \( x = -\frac{11}{4} \): \[ -2\left(-\frac{11}{4}\right) = \frac{22}{4} = \frac{11}{2} \quad (\text{this is correct}) \]
Thus, the solution \( x = -\frac{11}{4} \) corresponds to option d, \( -2x = \frac{11}{2} \).