Q.For the inequation $x+y\le 5$, the half-plane that forms its solution: (A) contains the origin and includes the line (B) contains the origin but excludes the line (C) does not contain the origin (D) is the line $x+y=5$ only

Test the origin and note the sign is weak ($\le$).

The boundary line is $x+y=5$. Because the sign is $\le$ (weak), the line is solid and included in the solution — this already rules out (B), which excludes the line.

Test the origin (the line does not pass through it): $0+0=0\le5$ is true, so the solution is the half-plane containing the origin. This rules out (C).

Option (D) is wrong because the solution is a whole half-plane (a region), not merely the boundary line.

Hence the correct description is (A): the half-plane containing the origin, together with the solid boundary line. …

Choosing (B) by reflexively drawing a dashed line — dashed is only for strict $<,>$; a weak …