If x > y and x < 0, then which of the following must be true?

We're told that x is NEGATIVE, and we're told that y < x, so y is also NEGATIVE

Take each statement and manipulate it (by applying the rules of inequalities) to see which one(s) obey the restriction that y < x

I. \(\frac{1}{x} < \frac{1}{y}\)

Multiply both sides by x to get: \(1 > \frac{x}{y}\) [since we multiplied by a NEGATIVE value, we REVERSED the inequality symbol]

Multiply both sides by y to get: \(y < x\) [since we multiplied by a NEGATIVE value, we REVERSED the inequality symbol]

PERFECT! This obeys the restriction that y < x

Statement I is true.
Check the answer choices . . . ELIMINATE B and C


II. \(\frac{1}{x−1} < \frac{1}{y−1}\)
NOTE: Since x is NEGATIVE, we know that x-1 is NEGATIVE
Likewise, since y is NEGATIVE, we know that y-1 is NEGATIVE

Multiply both sides by \(x-1\) to get: \(1 > \frac{x-1}{y−1}\) [since we multiplied by a NEGATIVE value, we REVERSED the inequality symbol]

Multiply both sides by \(y-1\) to get: \(y-1 < x-1\) [since we multiplied by a NEGATIVE value, we REVERSED the inequality symbol]

Add 1 to both sides to get: \(y < x\)

PERFECT! This obeys the restriction that y < x

Statement II is true.
Check the remaining answer choices . . . ELIMINATE A and E


By the process of elimination, the correct answer is D

RELATED VIDEO FROM MY COURSE

given x<0 ; x is -ve so y has to be -ve as well
x=-1 and y=-2
test with options
only IMO E is correct

The 3rd statement won't stand for x=-1, y=-2
In this case, it will be 1/x= 1/(-1+1)= infinity
While, 1/y= 1/(-2+1)=-1

Hence answer should be D, statement 1 and 2 are true

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