Solve the inequality 45x - 56 > 45/x

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The inequality `45x - 56 > 45/x` has to be solved.

`45x - 56 > 45/x`

=> `45x - 56 - 45/x > 0`

=> `(45x^2 - 56x - 45)/x > 0`

=> `(45x^2 - 81x + 25x - 45)/x > 0`

=> `(9x(5x - 9) + 5(5x - 9))/x > 0`

=> `((9x + 5)(5x - 9))/x > 0`

This is true when two of the terms is less than 0 and the third is greater than 0.

9x + 5 < 0 and 5x - 9 < 0 and x > 0

=> x < -5/9 and x < 9/5 and x > 0

This is not possible for any value of x

9x + 5 > 0 and 5x - 9 < 0 and x < 0

=> x > -5/9 and x < 9/5 and x < 0

=> `x in (-5/9, 0)`

9x + 5 < 0 and 5x - 9 > 0 and x < 0

=> x < -5/9 and x > 9/5 and x < 0

This is not possible for any value of x.

The solution of the inequality is (-5/9, 0)

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