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Quiz 1 Solutions

Quiz 1 Solutions - Version 57 1 Solve the inequality ln(x 2...

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Version 57 1. Solve the inequality ln( x - 2) + ln( x - 5) - ln( x - 4) < ln(3) . Give your answer in interval notation. Make sure that the equation actually makes sense for the interval you give in your answer. Solution: We exponentiate to get ( x - 2)( x - 5) ( x - 4) < 3 . We can assume that x - 4 > 0 for otherwise ln( x - 4) will not make sense. It is therefore OK to multiply by x - 4 . This gives ( x - 2)( x - 5) < 3( x - 4) or x 2 - 7 x + 10 < 3 x - 12 or x 2 - 10 x + 22 < 0 or ( x - 5) 2 - 25 + 22 < 0 or | x - 5 | 2 < 3 or 5 - 3 < x < 5 + 3 . But the original equation only makes sense for x > 5 and 5 - 3 < 5 , so the actual range is 5 < x < 5 + 3 . In interval notation (5 , 5 + 3) . 2. Find tan( x ) where x is the smallest solution of 3 tan( x ) + 7 tan x + π 4 = - 5 in the range - π 2 < x < π 2 . Solution: We have tan x + π 4 = tan( x ) + tan ( π 4 ) 1 - tan( x ) tan ( π 4 ) = tan( x ) + 1 1 - tan( x ) since tan π 4 = 1 . Therefore, setting t = tan( x ) we have 3 t + 7 t + 1 1 - t = - 5 which simplifies to 3 t (1 - t ) + 7( t + 1) = - 5(1 - t ) and 3 t - 3 t 2 + 7 t + 7 = - 5 + 5 t and 3 t 2 - 5 t - 12 = 0 and (3 t + 4)( t - 3) = 0 so that t = - 4 3 or t = 3 . But tan is increasing in the given interval, so the smaller value of x
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