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Quadratic Inequalities

Author:
November 5th, 2022

SOLVE FOR ‘X’ 9X^2 – 12X – 12 > 0

QUESTION :- SOLVE FOR X

1. 9X^2 -12X -12 >0

SOLUTION: The above given inequality is an quadratic which has two factors

9x^2 -12x-12 > 0

= 9x^2 –18x + 16x –12 > 0

= 9x(x-2) + 6(x-2) > 0

= (x -2) (9x + 6)>0

which implies the product of (x -2) & (9x + 6 ) > 0
Therefore two factors has to be +tive (or) –tive values
For x >2 we have ( x- 2) > 0 & ( 9x + 6 ) > 0 implies ( x -2 ) ( 9x + 6 ) > 0 ———-(1)

For x < – 6/9 we have ( x -2) < 0 & ( 9x + 6 ) < 0 implies ( x -2 ) ( 9x + 6 ) > 0 ——(2)

For –6/9 < x < 2 we get ( x -2 ) ( 9x + 6 ) < 0 —–(3)

From ( 1) (2) (3) we infer that ‘x’ belongs to [ ( -ά,- 6/9) U ( 2, ά) ] HENCE THE SOLUTION FOR ‘X’

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