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Bunuel
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If the quotient x/z is negative, which of the following must be true? 28 Oct 2024, 01:30
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If the quotient x/z is negative, which of the following must be true?

(A) x < 0

(B) z < 0

(C) x – z < 0

(D) x + z < 0

(E) xz < 0


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E
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satish_sahoo
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If the quotient x/z is negative, which of the following must be true? 28 Oct 2024, 04:02
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\(\frac{x}{z}\) can only be negative when either \(x\) or \(z\) will be negative at a time and not both. So, we know now that either \(x\) is negative or \(z\) is negative. Keeping this reasoning:

Since we don't know, which one of \(x\) or \(z\) is actually negative we can't determine if \(x<0\) or \(z<0\). hence, eliminate option A and B right away.

Notice, by the same logic, we can see scenarios where \(x-z\) can be true (or cannot) and \(x+z\) can be true (or cannot). Hence, both can't be MUST be true.

In E, \(xz\) will definitely be negative since, only one of these variables are negative making the product negative and hence, our quotient has to be negative.

Option E it is.

Hope it helps.
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If the quotient x/z is negative, which of the following must be true? 28 Oct 2024, 08:39
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But, if a denominator is negative and multiplied by negative quotient wouldn't it become positive ??
satish_sahoo
\(\frac{x}{z}\) can only be negative when either \(x\) or \(z\) will be negative at a time and not both. So, we know now that either \(x\) is negative or \(z\) is negative. Keeping this reasoning:

Since we don't know, which one of \(x\) or \(z\) is actually negative we can't determine if \(x<0\) or \(z<0\). hence, eliminate option A and B right away.

Notice, by the same logic, we can see scenarios where \(x-z\) can be true (or cannot) and \(x+z\) can be true (or cannot). Hence, both can't be MUST be true.

In E, \(xz\) will definitely be negative since, only one of these variables are negative making the product negative and hence, our quotient has to be negative.

Option E it is.

Hope it helps.
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If the quotient x/z is negative, which of the following must be true? 28 Oct 2024, 23:59
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Aman_KS
But, if a denominator is negative and multiplied by negative quotient wouldn't it become positive ??
satish_sahoo
\(\frac{x}{z}\) can only be negative when either \(x\) or \(z\) will be negative at a time and not both. So, we know now that either \(x\) is negative or \(z\) is negative. Keeping this reasoning:

Since we don't know, which one of \(x\) or \(z\) is actually negative we can't determine if \(x<0\) or \(z<0\). hence, eliminate option A and B right away.

Notice, by the same logic, we can see scenarios where \(x-z\) can be true (or cannot) and \(x+z\) can be true (or cannot). Hence, both can't be MUST be true.

In E, \(xz\) will definitely be negative since, only one of these variables are negative making the product negative and hence, our quotient has to be negative.

Option E it is.

Hope it helps.
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Hi, Not sure if I understood your query properly, but i'll try to elaborate here:

When we say that the quotient \(\frac{x}{z}\) is negative, we're not talking about a negative quotient multiplied by a negative denominator. Instead, we're saying that the division of x by z itself results in a negative value.

For a quotient to be negative:
  • One of x or z must be negative, and the other must be positive. This condition alone guarantees that the product \(xz\) will be negative.
We are simply trying to answer here, out of all the options which one has to be true all the time.

The key here is to figure out that the quotient \(\frac{x}{z}\) can only be negative when either one of the numerator/denominator is negative and if only one of them is negative then definitely their product is negative and hence, \(xz<0\).

  1. If x is positive and z is negative, the result \(\frac{x}{z}\) is negative.
  2. If x is negative and z is positive, the result \(\frac{x}{z}\) is still negative.
  3. If x and z both are positive / negative then the question falls apart coz then x/z can't be negative.

hence, E is the only MUST BE TRUE scenario.
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If the quotient x/z is negative, which of the following must be true? 29 Oct 2024, 00:50
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Option E.
Let x = -1, z = 2
a) x < 0; -1 < 0 "sure", but x can be any number. Therefore, this choice is very skeptical
b) z < 0; 2 < 0 "clearly not", but again y also can be any number.
c) x - z; -1 - 2 = -3 < 0, considering the choices above, this one is doubtful, too
d) x + z; -1 + 2 = 1 < 0, "no"
e) xz; -1 * 2 = -2, "correct answer"
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