Finding a set of maximally different solutions using linear programming or other optimization technique
A heuristic, using linear programming

One approach might be to pick a random objective function, and maximize it.  Then repeat, with a different set of objective functions each time.

In other words, suppose the unknowns are $x_1,x_2,\dots,x_n$, and you have some constraints $\mathcal{C}$.  In each iteration you pick $c_1,c_2,\dots,c_n \in \mathbb{R}$ randomly, then search for a solution that maximizes $c_1 x_1 + \dots + c_n x_n$ subject to the constraints $\mathcal{C}$.

I would expect this heuristic might often find a somewhat scattered set of solutions -- not necessarily maximally scattered (maximally far from each other), but probably not too close to each other, either.

Maximizing the average pairwise L2 distance, using quadratic programming

Alternatively, use quadratic programming.  For simplicity, let's look at the problem of finding two solutions.  Suppose you want two solutions $x,y$ that are as far apart from each other as possible, under the $L_2$ norm (Euclidean distance).  Then this can be formulated as a quadratic programming problem.

Basically, you want to maximize the squared distance $d(x,y)^2 = (x_1-y_1)^2+\cdots+(x_n-y_n)^2$ between $x$ and $y$, subject to the requirement that both $x$ and $y$ must satisfy the constraints.  This is the problem of maximizing a quadratic objective function, with linear constraints -- i.e., quadratic programming.

If you want $k$ points that are maximally scattered, this is also possible.  Say the points are $x^1,\dots,x^k \in \mathbb{R}^n$.  Then you could maximize the objective function

$$\sum_{i&lt;j} d(x^i,x^j)^2,$$

i.e., the function

$$\sum_{i&lt;j} \sum_\ell (x^i_\ell - x^j_\ell)^2.$$

This is a quadratic function, and you have linear constraints $\mathcal{C}$ on each of the points $x^i$, so this is a quadratic programming instance.  It finds you points that are maximally scattered in the sense that the average pairwise distance is maximized.

You can also formulate a greedy variant of this algorithm, where you already have $k$ solutions, and you want to find a $k+1$th solution that satisfies all the linear inequalities and also maximizes the average L2 distance from it to the other $k$ solutions.  That too can be formulated as a quadratic programming problem.

Quadratic programming is harder than linear programming, but there are off-the-self solvers that will solve quadratic programming problems for you.

Maximizing the minimal pairwise L2 distance, using QCQP

Finally, let's say you want your $k$ points to be scattered in the sense that you want to maximize the minimum pairwise distance.  In other words, let's say you want to find the largest possible threshold $t$ such that it is possible to find $k$ points $x^1,\dots,x^k \in \mathbb{R}^n$ that each satisfy the linear constraints, and such that each pair of points is at distance $t$ away from each other: $d(x^i,x^j) \ge t$ for all $i&lt;j$.  Then this can be formulated as a quadratic optimization program with quadratic constraints, i.e., QCQP.  QCQP is even harder, but there are off-the-shelf solvers for QCQP you could try out, too.