Which of the following inequalities have a finite range of values of x

A. X^2 + 5X >-6 you can see that X can be all positive including 0 so not finite
B. this shows X has to be >3 and <-7 so again not finite
C. move terms around gets 6x < 21 so x can be any negative value and couple of positives so infinite
D. x^2 - 4x <-3 You can readily see that X cant be negative since the -4X and x^2 results in positives thus invalidates the inequality. Try plugin couple of numbers - 0 doesnt work since 0<-3 is invalid. X=1 gets you -3<-3 so no. x=2 gets you 4-8 = -4<-3 which works x=3 gets 9-12=-3 <-3 so no. Stop here and you see x can be only 2

so D

Which of the following inequalities have a finite range of values of "x" satisfying them?
1.x^2 + 5x + 6 > 0
2.|x + 2| > 4
3.9x - 7 < 3x + 14
4.x^2 - 4x + 3 < 0

well i have seen that Q and answered correctly, but the suggestes solution for denying the 1st answer was:
Evaluating both the options, we get the range of values of "x" that satisfy this inequality to be x < -2 or x > -3. i.e., "x" does not lie between -2 and -3 or an infinite range of values.

after reading Fluke's, Karishma's and Bunuel's posts i thought inequalities are clearer.
though in my decent opinion that the answer should be x>-2 and x<-3.

correct me if i am wrong

Lets try and solve this one by one, dimri10.

1. x^2 + 5x + 6 >0
=> (x+2)*(x+3)>0
=> x>-2 or x<-3
You are right about the range of values here. The answer is x>-2 and x<-3. Lets test with a couple of values. Putting x=-4 (<-3) here we get a positive answer. Putting x=-1 (>-2) here we again get a positive answer. Putting x=-2.5 here we get a negative answer. Confirmed. The answer to this option is therefore not a finite range of values.

2. |x+2| > 4
=> x > 2 and x < -6
Again, not a finite range of values.

3. 9x - 7 < 3x + 14
=> x < 21/6
=> x < 7/2
A range, and not a finite range of values.

4. x^2 - 4x + 3 < 0
=> (x-1)*(x-3) < 0
=> x>1 and x<3
This is a finite range of values.

The answer is therefore (4).

for the first one the solution is x>-2 or x<-3 , not and. So we have two cases . if you consider either one of the case, its not yielding a finite range.

4th is the only option that can yield finite range solution.

hence 4th choice is correct.

4 alone ..
x^2 - 4x + 3 < 0
(x-1)*(x-3) < 0
x>1 and x<3

1. x^2 + 5x + 6 >0
=> (x+2)*(x+3)>0
=> x>-2 or x<-3

why x<-3? what is wrong with the below?
x+3>0
x>-3

I think you cannot divide that inequality by \((x+2)\) or \((x+3)\) to isolate \(x\). You don't know if they are positive or negative.

+1 D

Thanks @Karishma and @Gurupreet singh.........................Very useful post


It should be D

well explained by karishma . Thanks

I solved this by graphic approach like it has been suggested on other posts. you find the roots, draw the first two parabolas and x=21/6 and you realize that region isn't bound. The last one has roots 3 and 1, but is <0, so is bound.

Similar questions to practice:
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