Half-Planes & Regions

A Linear Inequality Is a Half-Plane

A linear inequality selects one side of its boundary line. This lesson shows how an easy test point chooses the side and how axis constraints clip the visible region. You will also see the greater-than case flip away from the origin.

highlighted = computed this step

Boundary line

The boundary for 2x+y<=4 is 2x+y=4. It meets the axes at (2, 0) and (0, 4). Why: changing from an inequality to its boundary shows the line that separates the allowed side from the rejected side.

2x+y=4intercepts (2,0) and (0,4)2x+y=4\quad \text{intercepts }(2, 0)\text{ and }(0, 4)
Half-plane choicesThe inequality side is clipped to the nonnegative axes for these first regions.Feasible region2x + y ≤ 4(0, 0)(0, 4)(2, 0)Feasible region2x + y ≥ 2(0, 2)(1, 0)

Origin side

Testing the origin gives 2 times 0 plus 0 equals 0, which is at most 4. So the origin side is feasible. Why: one easy test point identifies which side of the boundary the inequality keeps.

2⋅0+0=0≤42\cdot{}0+{}0=0\le{}4
Half-plane choicesThe inequality side is clipped to the nonnegative axes for these first regions.Feasible region2x + y ≤ 4(0, 0)(0, 4)(2, 0)Feasible region2x + y ≥ 2(0, 2)(1, 0)

Greater-than side

For 2x+y>=2, the same origin test gives 0>=2, which is false. The boundary intercepts are (1, 0) and (0, 2), and the feasible side is away from the origin. Why: the inequality direction, not the drawn line alone, decides the side.

2x+y=2intercepts (1,0) and (0,2)2x+y=2\quad \text{intercepts }(1, 0)\text{ and }(0, 2)
Half-plane choicesThe inequality side is clipped to the nonnegative axes for these first regions.Feasible region2x + y ≤ 4(0, 0)(0, 4)(2, 0)Feasible region2x + y ≥ 2(0, 2)(1, 0)

Clipped triangle

With the nonnegative axes, the visible region is the triangle with corners (0, 0), (2, 0), and (0, 4). The half-plane extends beyond the frame; shown clipped to the nonnegative axes. Why: adding axis constraints turns an open side of a line into the visible region used by the diagram.

corners (0,0),(2,0),(0,4)\text{corners }(0, 0),(2, 0),(0, 4)
Half-plane choicesThe inequality side is clipped to the nonnegative axes for these first regions.Feasible region2x + y ≤ 4(0, 0)(0, 4)(2, 0)Feasible region2x + y ≥ 2(0, 2)(1, 0)

Diagram note

A half-plane is one side of a boundary line; adding axes clips the visible part without changing the exact arithmetic. Pixel positions are rounded for layout; every number shown is exact.

boundary plus side test gives the half-plane\text{boundary plus side test gives the half-plane}
Half-plane choicesThe inequality side is clipped to the nonnegative axes for these first regions.Feasible region2x + y ≤ 4(0, 0)(0, 4)(2, 0)Feasible region2x + y ≥ 2(0, 2)(1, 0)