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
How many integrals value of \(x\) satisfy the inequality \((1-x^2)(4-x^2)(9-x^2) > 0 ?\)
This topic has expert replies
-
- Legendary Member
- Posts: 2214
- Joined: Fri Mar 02, 2018 2:22 pm
- Followed by:5 members
$$\left(1-x^2\right)\left(4-x^2\right)\left(9-x^2\right)>0$$
$$\left(1-x^2\right)=\left(1+x\right)\left(1-x\right)$$
$$\left(4-x^2\right)=\left(2+x\right)\left(2-x\right)$$
$$\left(9-x^2\right)=\left(3+x\right)\left(3-x\right)$$
$$Therefore,\ \left(1+x^2\right)\ \left(4-x^2\right)\ \left(9-x^2\right)>0$$
$$=>\ \left(1+x\right)\left(1-x\right)\left(2+x\right)\left(2-x\right)\left(3+x\right)\left(3-x\right)>0$$
This provides us with 3 range of x
$$1=x<-3$$
$$2=x>3$$
$$3=>x=0$$
$$if\ x<-3$$
$$\left(1-4\right)\left(1+4\right)\left(2-4\right)\left(2+4\right)\left(3-4\right)\left(3+4\right)<0$$
$$-3\cdot5\cdot-2\cdot6\cdot-1\cdot7<0$$
$$-15\cdot-12\cdot-7<0$$
$$-15\cdot-12\cdot-7<0$$
$$-1260<0$$
$$this\ does\ not\ satisfy\ the\ given\ \exp ression$$
$$if\ x>3$$
$$\left(1+4\right)\left(1-4\right)\left(2+4\right)\left(2-4\right)\left(3+4\right)\left(3-4\right)<0$$
$$5\cdot-3\cdot6\cdot-2\cdot7\cdot-1<0$$
$$-15\cdot-12\cdot-7<0$$
$$-1260<0$$
$$this\ does\ not\ satisfy\ the\ given\ inequality$$
$$if\ x=0$$
$$then\ \left(1+0\right)\left(1-0\right)\left(2+0\right)\left(2-0\right)\left(3+0\right)\left(3-0\right)>0$$
$$1\cdot1\cdot2\cdot2\cdot3\cdot3>0$$
$$36>0$$
$$this\ is\ the\ only\ integral\ value\ that\ satisfies\ the\ inequality$$
$$Since\ only\ 1\ integral\ value\ satisfies\ the\ inequality,$$
$$Answer\ =\ B$$
$$\left(1-x^2\right)=\left(1+x\right)\left(1-x\right)$$
$$\left(4-x^2\right)=\left(2+x\right)\left(2-x\right)$$
$$\left(9-x^2\right)=\left(3+x\right)\left(3-x\right)$$
$$Therefore,\ \left(1+x^2\right)\ \left(4-x^2\right)\ \left(9-x^2\right)>0$$
$$=>\ \left(1+x\right)\left(1-x\right)\left(2+x\right)\left(2-x\right)\left(3+x\right)\left(3-x\right)>0$$
This provides us with 3 range of x
$$1=x<-3$$
$$2=x>3$$
$$3=>x=0$$
$$if\ x<-3$$
$$\left(1-4\right)\left(1+4\right)\left(2-4\right)\left(2+4\right)\left(3-4\right)\left(3+4\right)<0$$
$$-3\cdot5\cdot-2\cdot6\cdot-1\cdot7<0$$
$$-15\cdot-12\cdot-7<0$$
$$-15\cdot-12\cdot-7<0$$
$$-1260<0$$
$$this\ does\ not\ satisfy\ the\ given\ \exp ression$$
$$if\ x>3$$
$$\left(1+4\right)\left(1-4\right)\left(2+4\right)\left(2-4\right)\left(3+4\right)\left(3-4\right)<0$$
$$5\cdot-3\cdot6\cdot-2\cdot7\cdot-1<0$$
$$-15\cdot-12\cdot-7<0$$
$$-1260<0$$
$$this\ does\ not\ satisfy\ the\ given\ inequality$$
$$if\ x=0$$
$$then\ \left(1+0\right)\left(1-0\right)\left(2+0\right)\left(2-0\right)\left(3+0\right)\left(3-0\right)>0$$
$$1\cdot1\cdot2\cdot2\cdot3\cdot3>0$$
$$36>0$$
$$this\ is\ the\ only\ integral\ value\ that\ satisfies\ the\ inequality$$
$$Since\ only\ 1\ integral\ value\ satisfies\ the\ inequality,$$
$$Answer\ =\ B$$
GMAT/MBA Expert
- Scott@TargetTestPrep
- GMAT Instructor
- Posts: 7249
- Joined: Sat Apr 25, 2015 10:56 am
- Location: Los Angeles, CA
- Thanked: 43 times
- Followed by:29 members
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
Founder and CEO
[email protected]
See why Target Test Prep is rated 5 out of 5 stars on BEAT the GMAT. Read our reviews