A vector space \(A\) over a field \(F\) is called an \(n\)-Lie algebra if it is endowed with an \(n\)-ary multilinear operation \([ ,\ldots ,\;]: A^n\rightarrow A\) which satisfies the generalized anticommutativity and Jacobi identities: \[ \begin{aligned} [x_1,\ldots,x_n] &=\text{sign} (\sigma)[x_{\sigma(1)},\ldots,x_{\sigma(n)}],\tag{i}\\ [x_1,\ldots,x_n]R(y) &=\sum_{i=1}^n[x_1,\ldots,x_iR(y),\ldots,x_n], \tag{ii} \end{aligned} \] where \(\sigma\in S_n\), \(R(y)=R(y_1,\ldots,y_{n-1})\) is the operator of right multiplication on \(A\) defined by \(R(y)(x)=[x,y_1,\ldots,y_{n-1}]\). This notion was introduced by \textit{V. T. Filippov} [Sib. Math. J. 26, 879-891 (1985); translation from Sib. Mat. Zh. 26, No. 6, 126-140 (1985; Zbl 0585.17002)]. There are various definitions of solvability for \(n\)-ary operations. The author defines an \(n\)-Lie algebra \(A\) to be \(L\)-solvable if the Lie algebra \(L(A)\) generated by all the operators \(R(y),\;y\in A^{n-1}\), is solvable. It is proven that, for a finite-dimensional \(n\)-Lie algebra \(A\) over a field of characteristic~0, the \(L\)-solvability condition is equivalent to any of the following two statements: a) for every representation \(\rho\) of \(A\), the Lie algebra \(L_{\rho}(A)\) generated by the set \(\rho(A)= \{\rho(y)\mid y\in A^{n-1}\}\) is solvable; b) if \(\rho\) is a finite-dimensional representation of \(A\) and the associative algebra \(A_{\rho}^*\) generated by the set \(\rho(A)\) is semisimple, then \(A^2\subseteq\ker\rho\). Finally, it is proven that if the Lie algebra \(L(A)\) is semisimple then \(A\) is decomposed into a direct sum of the center and simple ideals \(A_i\), with every \(L(A_i)\) semisimple and \(L\cong\bigoplus L(A_i)\).