Which statement satisfies both (x\le 6) and (x>6) at the same time?

Correct answer: no (x\in\mathbb{R})

The direct answer is A: there is no real number satisfying both conditions. The first inequality x≤6 allows 6 and every number smaller than 6. The second inequality x>6 allows only numbers strictly larger than 6. These two sets have no common element, because a number cannot be at most 6 and greater than 6 at the same time. Therefore their intersection is empty, written as ∅, or “no x∈R.” A is correct. B is wrong because x=6 satisfies x≤6 but fails x>6. C is wrong because numbers greater than 6 do not satisfy the first inequality, and numbers less than or equal to 6 do not satisfy the second. D is wrong because x>0 includes values such as 1, which fail x>6, and it still does not describe the common solution. On a number line, x≤6 shades left including 6, while x>6 shades right excluding 6; there is no overlap. Memory cue: opposite strict sides at the same boundary give an empty intersection.