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Find the greatest negative integer satisfying x2 − 4x − 77 < 0 and x2 > 4.

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x2 − 4x − 77 < 0 

⇒ x2 − 4x + 4 < 81

⇒ (x − 2)2 < 92 

⇒ −9 < x − 2 < + 9

⇒ − 7 < x < 11 ....(1)

The second inequality

x2 − 4 > 0 

⇒ x > 2 or x < − 2 .....(2)

The solution set satisfying both inequalities is the common part displayed in the diagram, namely, (− 7, − 2) ∪ (2, 11). Therefore, the greatest negative integer in this set is −3.

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