Solve the following inequality (x+1)/(x^4-x^3+64x-64)≥0.

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The inequality to be solved is `(x+1)/(x^4 - x^3 + 64x -64) >= 0` .

`(x+1)/(x^4 - x^3 + 64x -64) >= 0`

`(x+1)/((x-1)*(x+4)((x^2 - 4*x + 16)) >= 0`

(x^2 - 4*x + 16) is always greater than 0

The expression `(x+1)/((x-1)*(x+4))` is positive either when none of the terms are negative:

=> `(x+1) >= 0, (x-1) >= 0 and (x+4)>= 0`

=> `x >= -1, x >= 1 and x >= -4`

This true for all `x >= 1`

or when 2 of (x+1), (x-1) and (x+4) are negative

`(x+1) < 0, (x-1) < 0 and (x+4)>= 0`

=> `x < -1 and x < 1 and x >= -4`

This is true when x lies in [-4 , -1)

`x < -1 and x >= 1 and x < -4` which is not true for any x

`x >= -1 and x < 1 and x < -4` which is not true for any x

The inequality holds for [-4, -1)U(1, inf.)

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