Reformulate the problem as an LP
$$\min ||x||_1$$ subject to $$Ax = b$$ where $A \in \Bbb R^{mxn}, b \in \Bbb R^m$
and find the simplest (most compact form) LP dual by Lagrange duality and further simplification if necessary.
For the solution isn't this already an LP problem what does it mean by reformulating it as an LP?(constraint can be rewritten as $Ax - b = 0$)
A standard trick is to introduce variables $t_i$ and reformulate the optimization problem equivalently as \begin{align} \text{minimize} &\quad \sum_{i=1}^n t_i \\ \text{subject to} & \quad Ax = b, \\ & \quad t_i \geq x_i \quad \text{for } i = 1,\ldots, n, \\ & \quad t_i \geq -x_i \quad \text{for } i = 1,\ldots, n. \end{align} The variables in the reformulated problem are $x \in \mathbb R^n$ and the scalars $t_1, \ldots, t_n \in \mathbb R$.
Notice that $t_i \geq | x_i|$ if and only if $t_i \geq x_i$ and also $t_i \geq -x_i$.