I was trying to solve this inequality with two absolute values: | 2 x &#x2212;<!-- − -

enrotlavaec 2022-06-30 Answered
I was trying to solve this inequality with two absolute values:
| 2 x 3 | + 7 3 x 3 | x 7 |
I've got an empty set of solutions, but it's not correct.
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Answers (1)

Braylon Perez
Answered 2022-07-01 Author has 34 answers
The fastest way for me is to write down LHS-RHS as a piecewise linear function.
f ( x ) = | 2 x 3 | + 7 3 x + 3 | x 7 | = { 3 2 x + 7 3 x + 3 ( 7 x ) = 31 8 x , x 3 2 2 x 3 + 7 3 x + 3 ( 7 x ) = 25 4 x , 3 2 x 7 2 x 3 + 7 3 x + 3 ( x 7 ) = 2 x 17 , x 7
Note that f ( 3 2 ) = 19 , f ( 7 ) = 3
When x 3 2 , f is decreasing so f ( x ) > 0;
When x 7 , 2 x 17 0 x 17 2 .
Therefore 25 4 x 17 2 .
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