What type of region is given by the common graph of (y\leq x+4) and (y\geq x-2)?

Correct answer: Closed strip between two parallel lines

The direct answer is option A: a closed strip between two parallel lines. Rewrite the inequalities together: y≤x+4 and y≥x-2, so x-2≤y≤x+4. For every fixed x, y must lie between the lower value x-2 and the upper value x+4. The boundary lines y=x+4 and y=x-2 have the same slope, 1, but different intercepts, 4 and -2. Therefore they are distinct parallel lines and never meet. Because both signs include equality, both boundary lines are part of the solution. Hence the common region is the closed band or strip between them, extending infinitely in both directions. Option A is correct. Option B, the region outside both lines, is wrong because a point must lie below the upper line and above the lower line simultaneously, not outside both. Option C, only one line, is wrong because every point in the band between the lines is also included, not just the boundaries. Option D, empty region, is wrong because there are many solutions; for example, (0,0) satisfies 0≤4 and 0≥-2. The word closed is important: ≤ and ≥ include the boundary. If one sign were strict, the corresponding boundary would be excluded and the strip would not be closed on that side. Remember: same slope means parallel lines; between lower and upper bounds gives a strip.