inequality

[jainrahul1985]

If x is negative, is x < -3? (1) x^2 > 9 (2) x^3 < -9

OA A

[neelgandham]

If x<0, Is x < -3?

(1) x^2 > 9

Implies x > 3 or x < -3. The value of x cannot be greater than 3 because the question states that x is a negative number. We are now sure that the value of x is less than -3. Hence sufficient.

(2) x^3 < -9

x < (-9)^(1/3). The value of x is less than -2.XYZ .We cannot answer the question(Is x <-3) because the value of x can be between -2.XYZ and -3 or between -3 and Infinity(on the negative ). Hence Insufficient.

Answer A

[Anurag@Gurome]

If x is negative, is x < -3? (1) x^2 > 9 (2) x^3 < -9

OA A

(1) x² can be greater than 9 if x > 3 or x < -3 because 3² = 9 and (-3)² = 9. Since x is negative, so the only possible value of x = -3; SUFFICIENT.

(2) x^3 < -9
If x = -2.1, then x^3 = -9.2 < -9; here x > -3
If x = -4, then x^3 = -64 < -9; here x < -3
No definite answer; NOT sufficient.

The correct answer is A.

[amit2k9]

a x> -3 then only x^2 > 9. hence sufficient.

b x= -2.5 | -4 we get -12.25 | -64. hence not sufficient.

A it is.

[lunarpower]

i received a PM regarding this thread.

anurag --

(1) x² can be greater than 9 if x > 3 or x < -3 because 3² = 9 and (-3)² = 9. Since x is negative, so the only possible value of x = -3; SUFFICIENT.

this is a correct solution of x^2 > 9 --> x>3 or x<-3.
however, the second part ("the only possible value of x = -3") is incorrect; the possible values of x are all values for which x < -3, i.e., all points to the left of -3 on the number line.

(2) x^3 < -9

with ODD powers and roots -- even in inequalities -- you can just do the powers and roots; you don't have to worry about positives/negatives or double cases (as you would with quadratics and other even powers).
therefore,
x < -(cube root of 9)
the cube root of 9 is just barely more than 2 (because the cube root of 8 is exactly 2); let's say about 2.1.
so, this statement just means x < -2.1.
that's not good enough to establish whether x < -3; see the specific cases listed by other posters for concrete proof.

[Ian Stewart]

this is a correct solution of x^2 < 9 --> x>3 or x<-3.

You surely meant to write "x^2 > 9" here.

[lunarpower]

You surely meant to write "x^2 > 9" here.

sure did ... dang alligators, always turning around on me.

i edited that, so you can kill this post now if you want. thanks.