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Question

The number of real solutions of |x+3||4x|=|8+x| is

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Solution

There are 3 turning points here: 8,3,4
So we have 4 cases:
1) x<8
(x+3)(4x)=(8+x)x=1
We reject the solution because our condition is not satisfied (1 is not less than 8)
2)8x<3
(x+3)(4x)=(8+x)x=15
We reject the solution because our condition is not satisfied (15 is not within (8,3) interval.)
3)3x<4
(x+3)(4x)=(8+x)x=9
We reject the solution because our condition is not satisfied ( 9 is not within (3,4) interval.)
4) x4
(x+3)+(4x)=(8+x)x=1
We reject the solution because our condition is not satisfied (1 is not more than 4 )
Thus there is no solution.

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