calculate (|x| -3)(lnx+4)<0



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You need to remember that a product of two factors is negative if factors have different signs, one is positive and one is negative.

Considering the first case `|x|-3<0`  and `ln x+4>0`  yields:

`{(|x|-3lt0),(ln x+4gt0):} =gt {(|x|lt3),(ln xgt-4):}` `=gt {(-3ltxlt3),(xgte^(-4)):}` => `x in (-3,3)nn (1/e^4,oo)` `=> x in (1/e^4,3)`

Considering the second case `|x|-3>0`  and `ln x+4<0`  yields:

`{(|x|-3gt0),(ln x + 4 lt 0):} =gt {(|x|gt3),(ln x lt -4):}` => `{(-3gtxgt3),(xlt1/e^4):}` =>

`x in (-oo,-3)U(3,oo) nn (-oo,1/e^4)` => `x in (-oo,-3)U(3,oo) `

Hence, considering both cases, yields either `x in (1/e^4,3` ), if `|x|-3<0`  and `ln x+4>0` , or `x in (-oo,-3)U(3,oo), ` if `|x|-3>0`  and `ln x+4<0` .

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