Let A = x R : |x - a| 0 and d > 0 . If A B = (1, 2] [6, 7) , then the value of a + 2b + 3c + 4d is

Step-by-step solution

We are given A = x R : |x - a| 0 . -b x (a - b, a + b) We are also given B = x R : |x - c| d and d > 0 . x - c -d or x - c d x (- , c - d] [c + d, ) The intersection of sets A and B is: A B = (a - b, c - d] [c + d, a + b) We are given that A B = (1, 2] [6, 7) . Comparing the intervals, we get: a - b = 1 c - d = 2 c + d = 6 a + b = 7 Solving these equations: Adding the first and fourth equations: 2a = 8 a = 4 . Substituting a = 4 in the first equation: 4 - b = 1 b = 3 . Adding the second and third equations: 2c = 8