Is it possible to analytically calculate the size (i.e. the area) of the solution set for the following inequalities: $1<a<5$ and $1<b<5$ and $1<c<5$ and $1<d<5$ and $a\neq c.$ With $a,b,c,d$ the four variables.
For example for simpler inequalities, one sometimes gets lucky and the solution set is trivial: an example would be $x+y<4$ and $x>0$ and $y>0$ which is a triangle with base and height equal to $4$, so the size of the solution set is just the area of this.
You have a system of inequalities in four variables which would yield a four-dimensional polyhedron. So we do not call the size "area". The Lebesgue measure is the same as area when used in two dimensions but it applies to any number of dimensions, so let's call the size "measure" rather than "area".
The area of a rectangle is length times width. The volume of a rectangular prism (rectangular parallelepiped) is length time width times height. To extend this, the Lebesgue measure defines the measure of an n-dimensional right-angle prism as the product of all the side lengths. The length of the interval $[a, b]$ is $b-a$ if $b>=a$.
If we remove the limitation $a\ne c$ from your system we get a four-dimensional right-angle prism that exactly fits that previous paragraph, so the measure of that solution set is
$$(5-1)\cdot (5-1)\cdot (5-1)\cdot (5-1) = 4^4 = 256$$
The set $a=c$ defines a 3-dimensional hyperplane in 4-dimensional space which has a Lebesgue measure of zero. Putting the restriction $a\ne c$ back in your system removes a subset of that plane which will also have a measure of zero. So that restriction does not change the measure of your solution set, and your final answer is:
Of course, that means the number of solutions is infinite--the order of the continuum, to be precise.
One theorem in Lebesgue theory is that any set with finitely many or countably many points will have measure zero. Therefore, any solution set with non-zero measure will have infinitely many, in fact non-countably many, points.
In your example of $x>0,\ y>0,\ x+y<4$, the solution set is a triangle with base and height $4$ thus an area of $8$. So we immediately see that there are infinitely many solutions. We can easily list infinitely many of them,
$$x=1,\ y=1,\ 1.1,\ 1.01,\ 1,001,\ 1.0001,\ \ldots$$
But that is countably many solutions. To get uncountably many, set
$$x=1,\ y\in [0,\ 1]$$
The values of $y$ are literally the unit interval, thus uncountably many.
So we see that the number of solutions and the area of the solution set are two very different things, even if there are some relations between them. My answers assumed a little knowledge of modern set theory, talking about different kinds of infinity. But even if you do not know set theory you should see that counting the points and finding the area/measure of the points are two very different things.