If x is negative and the absolute value of 1/x is greater than 50, whi

Question

If x is negative and the absolute value of 1/x is greater than 50, which of the following about x must be true?

(A) x<-50
(B) x<-10
(C) -1<x<-1/2
(D) -1/50<x<0
(E) x>1/50

KarishmaB · Accepted Answer

Subbuc1991

Yes, you are right about (E).

1/|x| > 50

|x| < 1/50 (multiplying by |x| and dividing by 50 - both positive numbers)

This means -1/50 < x < 1/50 (Distance of x from 0 is 1/50)

But since x must be negative, only values lying in -1/50 < x < 0 satisfy.

Answer (D)

Abhishek009 · Answer

|\frac{1}{x}| > 50

Thus, x must be a fraction, (only (D) and (E) remains) , plug in and check the option, answer must be (D)

VikasGmail · Answer

Can you please explain the answer little bit more, my understanding is, if x is -ve then in this case 1 / x < - 50, and if we take reciprocals when the signs are opposite, the inequality does not change, so from that logic, it becomes x < - 1 / 50. Whereas in the answer D x > - 1 / 50. I am confused

Subbuc1991 · Answer

VeritasKarishma  Kindly help me understand why D is the answer. I eliminated E because it makes ' X ' positive whereas in the question it is mentioned that X is negative. Am I right in making that assumption. Kindly give your inputs.

nitsmash · Answer

X < 0 => X is Negative _________(i)

|1/X| > 50

1/X > 50 OR -1/X < 50

IN EQUALITIES FOR RECIPROCAL IF SAME SIGNS ON BOTH SIDES FLIP THE SIGNS IF NOT DON'T

X < 1/50 OR -X < 1/50

SINCE X IS NEGATIVE

THEREFORE the SECOND CASE APPLIES.

MULTIPLYING BOTH SIDES BY -1 AND FLIPPING THE SIGN

X > -1/50 _______(ii)

FROM (i) & (ii)

-1/50<X<0

huzaifahirani · Answer

So we know that |1/X| > 50
The solutions for the inequality is going to (-1/x) or (1/x)
They tell us that X is negative so we use the first case.
(-1/x) > 50
(-1) < 50(x)
(-1/50) < x

Since x is negative meaning less than zero
We get (-1/50) < x < 0

jamesp8403 · Answer

Just using some logic here, there's no way that D could possibly be the answer. If x were positive, it would have to be smaller than 1/50 and greater than zero. Since we know that x is less than zero, it has to be less than -1/50 and greater than -1.

Think about it. If x were greater than -1/50, say for example -1/40, then the absolute value would be 40, which is obviously not greater than 50.

Thinking about it algebraically, 1/-x > 50. Multiply both sides by -x and we have 1 < -50x. Divide both sides by -50 and we have -1/50 > x.
In other words x has to be less than -1/50.

Logical check here, think of something less than -1/50, like -1/60. The absolute value of -1/60 is 60. Checks out, that's greater than 50.

x can't be greater than -1/50 and less than zero. That is not possible.

D is not a viable answer.

Seriously, -1/49, absolute value is 49. 49 is greater than 50? Really?

I'm not sure what I'm missing here. The answer should be -1 < x < -1/50

KarishmaB · Answer

Imagine the number line:

............-2 ....................... -1 .....................0.......................1 ................. 2.......................

Now think about this: where does 1/2 (=0.5) lie? and where does 1/3 (=0.33) lie? I think you will agree with the following:

............-2 ....................... -1 .....................0.....1/3.....1/2.............1 ................. 2.......................

Now think about this: where does -1/2 (=-0.5) lie? and where does -1/3 (=-0.33) lie?

............-2 ....................... -1 ..........(-1/2)...(-1/3)........0.....1/3.....1/2.............1 .................

Do you see that -1/2 will be smaller than -1/3 because it is more negative.

Similarly, -1/40 is smaller than -1/50. Hence, -1/40 lies between -1 and -1/50.

On the other hand, -1/60 is greater than -1/50. Hence, -1/60 lies between -1/50 and 0.

Since -1/60 is acceptable to us, the range we are looking for is -1/50 to 0.

Answer (D)

jamesp8403 · Answer

Thanks for clearing things up for me. Still not sure why my equation always ends up -1/50 > x. Too much math and not enough sleep, the struggle is real

KarishmaB · Answer

Let me tell you why  jamesp8403

We are given that x is negative. So -x would become positive. Say if x = -2, -x = 2 which is positive.

When you multiply both sides of your inequality by -x, you are flipping the inequality sign but since -x is positive, you shouldn't do that.

1/-x > 50

becomes 1 > -50x

Now divide both sides by -50 to get -1/50 < x

Paras96 · Answer

IMO D

We can take, the reciprocal from the values in option D and verify.

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