We have the equation: \(x + x + 1 + x + 2 + x + \cdot \cdot \cdot+ x+6 = 14(x+6)\)
Simplify to: \(7x + 1 + 2 +3 + \cdot \cdot \cdot + 6 = 14x + 84\)
Furthur simplify the left-hand side: \(7x + 21 = 14x+84\)
Subtract 7x from both sides: \(21 = 7x + 84\)
Subtract 84 from both sides: \(7x = -63\)
Divide by 7: \(x = \color{brown}\boxed{-9}\)
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