On a graph, what does (x\leq 0) represent?

Correct answer: left half-plane including vertical axis

The direct answer is option B: the left half-plane including the vertical axis. The inequality \(x\leq0\) talks only about the x-coordinate. A point has x-coordinate 0 on the y-axis, x-coordinate less than 0 to the left of the y-axis, and x-coordinate greater than 0 to the right. Therefore all points on the left side and all points on the y-axis satisfy the inequality. The boundary is the vertical line \(x=0\), which is the y-axis. Because the sign is \(\leq\), the boundary is included, so it is drawn as a solid line. Option A, the right half-plane, corresponds mainly to \(x\geq0\), not \(x\leq0\). Option B is correct because it includes points with negative x and points with x equal to zero. Option C, the upper half-plane, is controlled by y-values and would relate to an inequality such as \(y\geq0\), not x. Option D, the lower half-plane, relates to \(y\leq0\), not x. For example, \((-2,5)\) satisfies the inequality because \(-2\leq0\), while \((3,-4)\) does not because \(3\leq0\) is false. The y-coordinate can be anything; only x matters. Memory cue: x controls left-right, y controls up-down; the equality sign includes the boundary.