The correct option is D 1
We know that
sgn x=⎧⎨⎩1 x>00 x=0−1 x<0
(i) x2−9>0
i.e., x∈(−∞,−3)∪(3,∞)
Then x2−5x+6=0
⇒(x−2)(x−3)=0⇒x=2,3
No solution as x∈(−∞,−3)∪(3,∞)
(ii) x2−9=0
i.e., x=−3,3
Then x2+6=0
No real roots.
(iii) x2−9<0
i.e., x∈(−3,3)
Then x2+5x+6=0
⇒(x+2)(x+3)=0⇒x=−2 [∵x∈(−3,3)]
∴ x=−2 is the only solution.
Hence, only one solution is possible.