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

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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]

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

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M7MBA wrote:
Wed Jun 24, 2020 5:53 am
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]

Solution:

We see that when x is 1, -1, 2, -2, 3, or -3, the value of the expression on the left hand side of the inequality will be 0, which means none of these integer values satisfy the inequality. Therefore, this leaves 0, an integer greater than 3, or an integer less than -3 as the only possible values of x. Let’s examine them.

If x = 0, we have (1)(4)(9) = 36, which is greater than 0.

If x > 3, we see that all the factors of the left hand side of the inequality will be negative. So the expression, as the product of these factors, will be also negative, which is not greater than 0.

Similarly, if x < -3, we see that all the factors of the left hand side of the inequality will be negative. So the expression, as the product of these factors, will also be negative, which is not greater than 0.

Therefore, there is only one integer value of x that satisfies the inequality, namely, 0.

Answer: B

Scott Woodbury-Stewart
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scott@targettestprep.com

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