What is the common part of (x\in(-4,8]) and (x\in[0,10)) on the number line?

Correct answer: ([0,8])

The direct answer is option B: \([0,8]\). The common part means the numbers that belong to both intervals at the same time. The first interval, \((-4,8]\), contains numbers greater than -4 and up to and including 8. The second interval, \([0,10)\), contains numbers from 0 including 0 to numbers less than 10. The overlap begins at 0 because the second interval includes 0 and 0 is also inside the first interval. The overlap ends at 8 because the first interval includes 8 and 8 is also inside the second interval. Thus the intersection is \([0,8]\). Option A, \((-4,10)\), is too large because it includes numbers such as -3 and 9 that are not in both intervals. Option B includes exactly the common part. Option C, \((0,8)\), wrongly excludes both endpoints, although 0 and 8 are included. Option D, \([-4,8]\), wrongly includes values from -4 to 0 that are not in the second interval; -4 itself is also excluded from the first interval. Memory cue: for intersection, take the larger left endpoint and the smaller right endpoint, carefully checking brackets.