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The equation \(x = |x-|30-2x||\) is equivalent to\(x = |x-2|x-15||\)
If \(x ≥ 15\), then \(x = |x-2|x-15||\) or \(x = | x – 2(x-15) | = | x – 2x + 30 | = | -x + 30 | = | x – 30 |\)
If \(x ≥ 30\), then \(x = | x – 30 | = x – 30\) or \(0 = -30\), which doesn’t make sense.
If \(15 ≤ x < 30,\) then \(x = - ( x – 30 ) = -x + 30\) or \(2x = 30.\) It follows that \(x = 15.\)
If \(x < 15\), then \(x = |x-2|x-15||\) is equivalent to \(x = | x + 2(x-15) | = | x + 2x - 30 | = | 3x - 30 | = 3| x – 10 |\)
If \(10 ≤ x < 15\), then \(x = 3| x – 10 | = 3(x-10) = 3x -30,\) so, \(2x – 30 = 0.\) It follows that \(x = 15,\) which is not a solution since \(10 ≤ x < 15.\)
If \(x < 10,\) then \(x = 3| x – 10 | = -3(x-10) = -3x + 30\) and \(4x = 30.\)
So, \(x = 7.5.\)
Thus, there are two solutions: \(7.5\) and \(15.\)
Therefore, the answer is C.
Answer: C
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