Confused in how to insert a slack variable in a constrain inequality

According to my understanding , we should put a slack variable to equate an inequality constrain by inserting the slack varaible in the side that is less than the other side , for example if we have 4x+2<2 this will be 4x+2+slack_variable=2 .But here in Wikipedia : In the example section , it says the following “By introducing the slack variable y>=0, the inequality Ax<=b can be converted to the equation y-Ax+b = 0” Which means that the slack variable is inserted in the bigger side ! Please some one explain this confusion.

Does the isometry group of a closed simple smooth curve in the plane constrain its perimeter^2/area ratio?

Let $ C$ be a simple closed smooth curve delimitating a bounded domain $ D$ in the euclidean plane of isometry group $ G$ and of given area $ A$ . Does the minimal possible ratio $ \dfrac{P^{2}}{A}$ where $ P$ is the perimeter hence the total length of $ C$ decrease when $ G$ runs over a sequence $ (G_{i})_{i>0}$ of groups such that $ i<j$ implies $ G_{i}$ is a strict subgroup of $ G_{j}$ ?

Linear programming IFF with equality constrain

Is it possible to write the following logical constrain in linear programming?

Let $ v$ be an integer variable and $ k$ an integer constant. Let $ y$ be a binary variable. The logical constrain requires:

$ y=1 \Longleftrightarrow v=k$

I need this kind of contrain in linear programming to use it in AMPL, but I really can’t find a way to write it down as a linear constrain.

Thanks to who’ll help me!