Problem Solving

How many integrals value of \(x\) satisfy the inequality \((1-x^2)(4-x^2)(9-x^2) > 0 ?\)

A. 0
B. 1
C. 3
D. 5
E. Greater than 5

[spoiler]OA=B[/spoiler]

Source: e-GMAT

$$\left(1-x^2\right)\left(4-x^2\right)\left(9-x^2\right)>0$$
$$\left(1-x^2\right)=\left(1+x\right)\left(1-x\right)$$
$$\left(4-x^2\right)=\left(2+x\right)\left(2-x\right)$$
$$\left(9-x^2\right)=\left(3+x\right)\left(3-x\right)$$
$$Therefore,\ \left(1+x^2\right)\ \left(4-x^2\right)\ \left(9-x^2\right)>0$$
$$=>\ \left(1+x\right)\left(1-x\right)\left(2+x\right)\left(2-x\right)\left(3+x\right)\left(3-x\right)>0$$
This provides us with 3 range of x
$$1=x<-3$$
$$2=x>3$$
$$3=>x=0$$
$$if\ x<-3$$
$$\left(1-4\right)\left(1+4\right)\left(2-4\right)\left(2+4\right)\left(3-4\right)\left(3+4\right)<0$$
$$-3\cdot5\cdot-2\cdot6\cdot-1\cdot7<0$$
$$-15\cdot-12\cdot-7<0$$
$$-15\cdot-12\cdot-7<0$$
$$-1260<0$$
$$this\ does\ not\ satisfy\ the\ given\ \exp ression$$
$$if\ x>3$$
$$\left(1+4\right)\left(1-4\right)\left(2+4\right)\left(2-4\right)\left(3+4\right)\left(3-4\right)<0$$
$$5\cdot-3\cdot6\cdot-2\cdot7\cdot-1<0$$
$$-15\cdot-12\cdot-7<0$$
$$-1260<0$$
$$this\ does\ not\ satisfy\ the\ given\ inequality$$
$$if\ x=0$$
$$then\ \left(1+0\right)\left(1-0\right)\left(2+0\right)\left(2-0\right)\left(3+0\right)\left(3-0\right)>0$$
$$1\cdot1\cdot2\cdot2\cdot3\cdot3>0$$
$$36>0$$
$$this\ is\ the\ only\ integral\ value\ that\ satisfies\ the\ inequality$$
$$Since\ only\ 1\ integral\ value\ satisfies\ the\ inequality,$$
$$Answer\ =\ B$$