For any real numbers a (a ≠ 0) and B, the inequality | a + B | + | A-B | ≥ m · | a | holds. Note that the maximum value of real number m is m. (1) find the value of M; (2) solve the inequality | X-1 | + | X-2 | ≤ M

For any real numbers a (a ≠ 0) and B, the inequality | a + B | + | A-B | ≥ m · | a | holds. Note that the maximum value of real number m is m. (1) find the value of M; (2) solve the inequality | X-1 | + | X-2 | ≤ M

(1) The inequality | a + B | + | A-B | ≥ m ·| a |, that is, m ≤| a + B | + | a − B | a |, holds for any real number a (a ≠ 0) and B |, so long as the left constant is less than or equal to the minimum value on the right (2 points) because |a + B + b|a-b | (a + b) + (a-b) | = 2 A, if and only if (a-b) (a + b) (a + b) = (a-b) (a + b) (a + b) ≥ 0 is set, that is, when |a + B; (a + B + A + B + B | (a + B + B + |a + |a |a + B + ? + ??a + ?a M = 2 (5 points) (2) inequality | X-1 | + | X-2 | ≤ m, that is | X-1 | + | X-2 | ≤ 2. Since | X-1 | + | X-2 | represents the sum of the distances from the corresponding points of X on the number axis to the corresponding points of 1 and 2, and the sum of the distances from the corresponding points of 12 and 52 on the number axis to the corresponding points of 1 and 2 is exactly equal to 2, the solution set of | X-1 | + | X-2 | ≤ 2 is: {x | 12 ≤ x ≤ 52}. (10 points)