a half-plane
A straight line divides the flat plane into two pieces, like a long fence splitting a field into 'this side' and 'that side'. Each of those two regions is a half-plane. While an equation like 2x + y = 4 picks out a one-dimensional line, an inequality like 2x + y < 4 picks out a whole two-dimensional region — one of the two halves.
Here is how it works. The line A x + B y = C is the boundary. Replacing the equals sign with an inequality, A x + B y < C or A x + B y > C, selects all the points on one side of that line: the expression A x + B y is smaller than C on one side and larger on the other. To decide which side an inequality names, just test a convenient point not on the line — the origin (0, 0) is easiest when the line does not pass through it. If the test point makes the inequality true, its whole side is the solution region; if false, the other side is. A strict inequality (< or >) excludes the boundary line itself (drawn dashed); a non-strict one (<= or >=) includes it (drawn solid).
Half-planes are the building blocks of linear programming and of describing regions: the overlap of several half-planes (the points satisfying several linear inequalities at once) carves out a polygon-shaped feasible region, the home of optimization problems in economics and operations. The one thing to get right is the test-point step — never assume the inequality points 'toward' or 'away from' the origin; check it. And remember that multiplying or dividing an inequality by a negative number flips its direction, a slip that silently sends you to the wrong half.
Which side does x + 2y >= 4 describe? The boundary is x + 2y = 4 (drawn solid, since >= includes it). Test the origin: 0 + 0 = 0, which is not >= 4, so the origin's side is excluded; the solution is the half-plane on the opposite side of the line from (0, 0).
Test the origin: if it fails the inequality, shade the other side; >= keeps the boundary line.
Do not guess the side from the inequality's direction — always test a point. And when you multiply or divide an inequality by a negative number, reverse the inequality sign, or you will name the wrong half-plane.