Algebra (inequality)

[asishkm]

Hi,

The direction is to select all options that are true

Q. If x^5 + x^2 < 0, then which of the following must be true?
1. x < -1
2. x < 0
3. x > 0
4. x > 1
5. x^4 < x^2

I solved the inequality:

x^2(x^3 + 1) < 0
=> x^3 + 1 < 0
=> x^3 < -1
=> x < -1

So that would mean both 1 and 2 are correct.

However the solution given at the back is only 1.

My thought is that since x must be less than -1, it is also obviously less than 0.

Thanks.

[xerocoool]

Hey,

You have solved it right,

try testing values

x^5 + x^2 < 0

=> simplified equation after solving x < -1

If x = -2
(-2)^5 + (-2)^2 < 0
-32 + 4 < 0
-28 < 0 => inequality holds true

If x = - 0.5 (that is x < 0)
(-0.5)^5 + (-0.5)^2 < 0
-0.03125 + 0.25 < 0
0.21875 < 0 (inequality does not hold true)

So x has to be less than -1

[tommywallach]

Or, to think about this more logically, if you know that x is less than -1, you also know that x is less than 0. (Just as you know that x is less than 142.)

Hope that makes sense!

-t

[asishkm]

Thanks for the reply.

Does it mean the answer (in a multiple answer correct question) would be the first two choices?

While for a single answer question, it would be the first choice.

I hope I got it right.

[xerocoool]

Hey Asish,

This inequality "x^5 + x^2 < 0" will never be true for any value between 0 and -1. But will hold true for all values less than -1 (i.e. -1.000001 to - infinity)

Even -1 fails (-1+1<0 : 0<0 has no meaning)

My thought is that since x must be less than -1, it is also obviously less than 0.

I think your interpretation of "x<-1" is slightly different. If x<-1 it does not imply that the equation will be true for values -1<=x<=0.

1) yes, if X < -1 we can conclude X has to be less than 0 (in general sense)
2) But with respect to the the inequality "x^5 + x^2 < 0" X is strictly less than -1

So even if it's a multiple choice question the answer will be only x<-1

[tommywallach]

Whoa whoa whoa. This is incorrect people.

If you know that x < -1, then you also know that x < 0. This is pure logic. If the question does not support that, then it has been written incorrectly. The discussion of plugging in values is irrelevant. It doesn't matter that numbers between 0 and -1 don't "work" in the equation. That doesn't matter. The point is that the answers that DO work must be less than zero. Your original issue with the question is valid.

-t