Which of the following is a value of x for which

Question

Which of the following is a value of x for which  x^{-9} –x^{-7} > 0 ?

A. -2
B. -1
C.  -\frac{1}{2} 
D. 1
E. 2

GMATGuruNY · Accepted Answer

Adding  x^{-7}  to both sides, we get:
 x^{-9} > x^{-7}

Since the resulting inequality is valid only if x is NONZERO, we can safely multiply both sides by   x^{10} , which must be a positive value:
 x^{10} * x^{-9} > x^{-7} * x^{10} 
 x > x^3

Scanning the answer choices, we can quickly see that -2 is a valid option for x:
 -2 > (-2)^3 
 -2 > -8

A

gmatophobia · Answer

x^{-9} –x^{-7} > 0 x^{-9}(1 –x^{2}) > 0 Case 1 : x > 0 If x > 0 → 1 - x^2 > 0 1 - x^2 > 0 x^2 < 1 We can eliminate options D, and E as in both the options x > 0 and x^2 \geq 1 Case 2 : x < 0 If x < 0 → 1 - x^2 < 0 1 - x^2 < 0 x^2 > 1 Options A, B, and C have values of x < 0 . However, among the three options, only in option A the value of x^2 > 1 . Hence, Option A is correct.

JeffTargetTestPrep · Answer

Using the exponent rules, we have: 1/x^9 – 1/x^7 > 0 We see that x cannot be equal to zero. If x > 0, then: 1 - x^9/x^7 > 0 1 - x^2 > 0 1 > x^2 |x| < 1 => -1 < x < 1 So, in this case, it must be true that 0 < x < 1 . If x < 0, then: 1 - x^9/x^7 < 0 |x| > 1 => x > 1 OR x < -1 So, in this case, it must be true that x < -1 . Therefore, x < -1 OR 0 < x < 1 . Answer: A

flyinghirsch · Answer

--- SIMPLE SOLUTION ---

Even though it does not look like it, the question is basically about neg * pos * neg:

Step 1: Factor out X^-9
X^-9 * (1 - x^2) > 0

Step 2: Difference of Square formula
X^-9*(1 - x)*(1+x) > 0

Step 3: Test cases 
Essentially product of 3 factors must be pos: 
For B) and D) solution of 3 factors would be 0 -> eliminate
For A) equation become (1/-2^9) * ( 1--2) * (1-2) -> sufficient as neg * pos * neg is positive and therefore > 0

FlyPig · Answer

x^-9 - x^-7 > 0
x^-9 * (1 - x^2) > 0
x^-9 * (1 - x) * (1 + x) > 0
We get the roots as 0, 1, or -1.

The critical points 0, 1, and -1 divide the number line into four intervals as shown. Let’s select a test point from each interval to determine the sign of the inequality within that interval, then we will obtain the answer.

Engineer1 · Answer

Please help. What is wrong with the way I am doing it? I am stuck after the case II in the paper solution below. How do I interpret this solution in terms of answer choices? Any feedback will help. Thank you. chetan2u KarishmaB

chetan2u · Answer

x^{-7}>0……\frac{1}{x^7}>0…..x^7>0

You are going wrong in changing the inequality sign. 
You do not change the sign of power and inequality sign. The negative powers just take the reciprocal but have no contribution in change of sign except that even powers are always positive.

kevinhirose · Answer

Can someone or @JeffTargetTestPrep explain how you went from 1/x^9 - 1/x^7 > 0 to 1 - x^2 > 0 ? I'm lost here.

Regor60 · Answer

X^-9 = 1/X^9 and X^-7=1/X^7 so
the problem can be restated as:

1/X^9 - 1/X^7 > 0

Multiply both sides by X^9:

X^9(1/X^9 - 1/X^7) > 0×X^9

= 1 - X9/X^7 > 0 =

1-X^2 > 0

Posted from my mobile device

naveeng15 · Answer

if we substitute the answers why cant we arrive at the answer ?

the value of x satisfies both -2 & -1/2?? 
which is the correct answer??
 KarishmaB   GMATNinja   bb   Bunuel

Bunuel · Answer

The correct answer is given under the spoiler in the original post and it's A, -2. -1/2 does not satisfy x^{(-9)} - x^{(-7)} > 0 : (-\frac{1}{2})^{(-9)} - (-\frac{1}{2})^{(-7)} = =(-2)^9 - (-2)^7 = =-512 + 128 = =-384 P.S. As explained above x^{(-9)} - x^{(-7)} > 0 holds for x < -1 and 0 < x < 1.

iamsujatha · Answer

look - thumb rule is that 2^9 is > 2^7, but in case of decimal, 1/2^9 < 1/2^7. Now as for the 1st option i.e. -2, substitution gives us -(1/2^9) + 1/2^7 i.e, 1/2^7 - 1/2^9, since we know that 1/2^7 > 1/2^9, the answer will be >0

HarshavardhanR · Answer

https://gmatclub.com/forum/download/file.php?id=83423 One way to solve this. Harsha GMAT-Club-Forum-6jlaqmff.png

KarishmaB · Answer

x^{-9} –x^{-7} > 0 Take x^-9 common x^{-9} (1 - x^2) > 0 \frac{(1-x^2)}{x^9} > 0 Now, x cannot be 1 or -1 because it makes the LHS 0. If x = -2, both numerator and denominator of LHS are negative so the fraction becomes positive. Hence this is correct. Note that for both 2 and - 1/2, the LHS will be negative. Answer (A) Method 2: If you want to solve it using the wavy line method, \frac{(x^2-1)}{x^9} < 0 \frac{(x +1)(x-1)}{x^9} < 0 Transition points are -1, 0 and 1 so the expression will be negative when x is between 0 and 1 OR less than -1. Only option (A) works. Wavy line method is discussed here: https://youtu.be/PWsUOe77__E

lucky21 · Answer

x^-9 - x^-7>0
1/x^9 = 1/x^7
x^7>X^9

a) -2^7 > -2^9 = true
b) -1^7 > -1^9 = -1>-1 (not true)
c) -1/2^7 >-1/2^9 = -1/128 > - 1/512 = 1/218 >1/512 (not true)
d) 1^7 > 1^9 = not true
e) 2^7 > 1^9 = not true
Ans: Option A

Usernamevisible · Answer

We need:

x^(-9) − x^(-7) > 0

Rewrite negative exponents:

1/x^9 − 1/x^7 > 0

Take common factor:

(1/x^9)(1 − x^2) > 0

So:

(1 − x^2)/x^9 > 0

Now test choices quickly.

A. x = −2

(1 − 4)/negative = negative/negative = positive

Works.

---

B. x = −1

1 − 1 = 0

Not > 0.

---

C. x = −1/2

1 − 1/4 = positive

x^9 is negative

positive/negative = negative

Does not work.

---

D. x = 1

1 − 1 = 0

Not > 0.

---

E. x = 2

1 − 4 = negative

x^9 positive

negative/positive = negative

Does not work.

Answer:

A. −2

Rickooreo · Answer

FAST METHOD

Given:

x^(-9) - x^(-7) > 0

Factor out the smaller power:

x^(-9)(1 - x^2) > 0

Now check the signs.

For x = -2:

• x^(-9) → negative (odd negative power)
• 1 - x^2 = 1 - 4 = -3 → negative
• Negative × Negative = Positive ✓

Therefore, A. -2 works.

NO CALCULATION NEEDED.

SIGN LOGIC:

For |x| > 1:
1 - x^2 is negative.

• x < -1 → x^(-9) is negative
 Negative × Negative = Positive ✓

• x > 1 → x^(-9) is positive
 Positive × Negative = Negative ✗

CHECK OPTIONS:

A. -2 → ✓
B. -1 → 0
C. -1/2 → ✗
D. 1 → 0
E. 2 → ✗

ANSWER: A. -2

EXAM TRICK:
For multiple-choice inequality questions, don't calculate the actual value.
Factor and determine the SIGN of each part.

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