maihuna wrote:If xyz ≠0, is x (y + z) >= 0?
(1) ¦y + z¦ = ¦y¦ + ¦z¦
(2) ¦x + y¦ = ¦x¦ + ¦y¦
Hi maihuna,
Always start by focussing on the stem and thinking about what is necessary to answer the question. The question tells us that none of x, y or z are zero. The question is asking whether x (y + z) > = 0. But the question is essentially asking whether x(y + z) is positive (many DS questions, especially yes/no questions can be rephrased).
In order for the product of two numbers to be positive, either they both have to be positive or else they both have to be negative. Accordingly, in order to answer this question we need to know whether the sign of x is the same as the sign of (y + z).
Statement One: |y + z| = |y| + |z|
Because there is no info about x, this statement is not sufficient. (Regardless of what (y + z)'s sign is, the answer to the question changes if x's sign changes, and we don't have info about x).
Statement Two: |x + y| = |x| + |y|
Because there is no info about z, this statement is not sufficient. (If x is a "large" negative number, then (x + y) is negative (-100 + 5). But if they are both positive, then (x + y) is positive).
Combo:
Let's consider the first statement first.
|y + z| = |y| + |z|
This means that y and z share the same sign.
Let's say the signs were different. What if y were positive and z negative? In algebra, we do operations within absolute value bars (and brackets) first before doing anything else. So, you would have:
Y - z = y + z
And if y were negative and z positive, you would have:
z - y = z + y
Given that none of the variables are zero, those two equations are clearly impossible.
One can also pick some numbers to help in seeing this.
Suppose y = 5 and z = -10 (different signs)
Then, |y| = |5| = 5 and |z| = |-10| = 10
and
|y| + |z| = |5| + |-10| = 5 +10 = 15
This is what the right hand side of the equation in statement 1 would be.
But |y + z| = |5 + (-10)| = |5 -10| = |-5| = 5
And that is what the left hand side would be.
Because 5 does not equal 15, if the signs of y and z were different, the equation in the statement would not be satisfied.
If you wanted to, you could use the same process of picking numbers to confirm that the equation in statement one is true so long as both y and z are positive or both negative. That is, you can now try 5, 10, and then -5, -10. Of course, you likely wouldn't have to.
By similar reasoning, we can see that statement two is telling us that the signs of x and y are the same.
So statement one tells us that y and z share the same sign. Statement two tells us that y and x share the same sign. Y is the variable that shows up in both equations. But because the statements can never contradict each other, Y's sign can't be different in the two statements. Combined, this means that all three of them (x, y, and z) share the same sign.
For example: if, in statement one, y is positive, z is also positive. But if y is positive in statement one, y is also positive in statement 2 (the statements cannot contradict). And, because y and x share the same sign in statement two, if y is positive, that means x (in addition to z) is also positive. Likewise, if y is negative, they will all be negative. Either they are all positive or else they are all negative.
Is x(y + z) positive?
Well, if all of them are positive we would have:
Pos*(pos + pos), and so the expression is definitely positive.
And, if all of them were negative, we would have:
Neg*(neg + neg) =
Neg* (neg) (sum of two negative numbers is always negative)
In this case, the expression is also clearly positive.
Because in both possible cases the answer to the question is "yes", the statements, although insufficient in isolation, are sufficient in combination.
Choose C
