Is x

Question

Is x < y ?

(1) z < y
(2) z < x

Its is clear that alone a or b is not sufficient to answer the question

But can't we combine and perform subtraction OR we have to apply the rule that we don't know the sign of z so we can't cancel it.Pls let me know the exact reason.

fluke · Answer

Sometimes, pure substitution is the fastest way to solve inequality. Please try with these:
z=0; x=1; y=2; x<y
OR
z=0; x=2; y=1; x>y
OR
z=0; x=1; y=1; x=y

ravsg · Answer

had one of the conditions been greater than and the other less than- combining both would have helped.

Posted from my mobile device

KarishmaB · Answer

You cannot subtract inequalities if they have the same sign. You can only add them in that case. 
You can subtract them only if they have opposite signs though I always suggest to flip the sign of one of them and then add them.

Case 1:
a < b
c < d
Add: a + c < b + d (Correct)
Subtract: a - c < b - d (Incorrect)

Case 2:
a < b 
c > d
Subtract: a - c < b - d (Correct)

Better yet, flip the sign and add:
a < b
d < c
Add: a + d < b + c (So that there is no confusion)

Note that either ways, you get the same resultant inequality

Another thing,
If you have a + b < a + c, you can cancel 'a' out irrespective of whether you know its sign or not. a is added/subtracted on both sides. Addition/subtraction of any number on both sides does not change the inequality.

Instead, if you have ab < ac, now you cannot cancel 'a' since you do not know its sign. a is multiplied/divided in this case. Multiplication/division by a negative number on both sides flips the sign of the inequality.

subhashghosh · Answer

Let z = 4, x = 5, y = 6 or z = 4, x = 7, y = 6

So in (1) and (2), x and y can be greater than each other by simply having the above stated values of x and y. Answer - E.

IanStewart · Answer

I'm concerned that your usage of the word 'sign' here might be confusing, since you are not talking about 'signs' (positives and negatives) at all; inequalities don't have 'signs'. You mean to say 'direction', surely.

In general it is not mathematically correct to subtract one inequality from another if they face in the same direction. You can learn rules for how to subtract inequalities, but those rules are both confusing and pointless, since you can get the same result by adding your inequalities. You *can* add two inequalities which face the same way, and in situations which resemble '2 equations/2 unknowns' problems, this can sometimes be useful. For example, if we know

x + y > 3
x - y > 1

then by adding the two inequalities (which are pointing in the same direction, so we can do this), we learn that 2x > 4, or that x > 2.

bumpbot · Answer

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Is x < y ? (1) z < y (2) z < x

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