If |x|

Question

If |x| < y, then which of the following must be true?

I. x < y
II. x < |y|
III. x + y > 0

A. I only
B. II only
C. III only
D. I & III only
E. I, II & III

Bunuel · Answer

|x| < y implies that y must be a positive number (since it's more than some non-negative value). Thus, |y| = y and I and II mean the same thing. Both must be true, because if y were less than x, we'd have 0 < y < x, and the absolute value of x would then be greater than y.

As for III, it must also be true, because if x is positive, then we'd have x + y = positive + positive = positive and if x is negative, then |x| = -x, and from the stem we'd get -x < y, which gives the same x + y > 0.

Answer: E.

BrushMyQuant · Answer

Given that |x| < y and we need to find which of the following must be true

We know that modulus of a number is always non-negative
=> y > |x|
=> y > 0

Now, there can be three cases. For something to be true it has to be true in all three cases
-Case 1: x < 0 I. x < y If x < 0, then x < y as y > 0 Ex: x = -2, y = 3 => -2 < 3 => TRUE II. x < \|y\| If x < 0 then \|y\| will be > x as \|y\|> 0 and x < 0 Ex: x = -2, y = 3 => \|3\| > -2 => TRUE III. x + y > 0 Now, x < 0 and y > 0 and y > \|x\| => x + y > 0 Ex: x = -2, y = 3, x + y = -2 + 3 = 1 > 0 => TRUE	-Case 2: x = 0 I. x < y If x = 0, then x < y as y > 0 Ex: x = 0, y = 2 => 0 < 2 => TRUE II. x < \|y\| If x = 0, then \|y\| will be > x as \|y\|> 0 and x = 0 Ex: x = 0, y = 2 => 0 < \|2\| => TRUE III. x + y > 0 Now, x = 0 and y > 0 => x + y > 0 Ex: x = 0, y = 2, x + y = 0 + 2 = 2 > 0 => TRUE	-Case 3: x > 0 I. x < y If x > 0, then x < y as y > \|x\| => y > x Ex: x = 2, y = 3 => 2 < 3 => TRUE II. x < \|y\| If x > 0, then \|y\| will be > x as y > \|x\| Ex: x = 2, y = 3 => 2 < \|3\| => TRUE III. x + y > 0 Now, x > 0 and y > 0 => x + y > 0 Ex: x = 2, y = 3, x + y = 2 + 3 = 5 > 0 => TRUE

=> All three are true in all the cases

So, Answer will be E
Hope it helps!

Watch the following video to MASTER Absolute Values

bumpbot · Answer

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Now, there can be three cases. For something to be true it has to be true in all three cases
-Case 1: x < 0 I. x < y If x < 0, then x < y as y > 0 Ex: x = -2, y = 3 => -2 < 3 => TRUE II. x < \|y\| If x < 0 then \|y\| will be > x as \|y\|> 0 and x < 0 Ex: x = -2, y = 3 => \|3\| > -2 => TRUE III. x + y > 0 Now, x < 0 and y > 0 and y > \|x\| => x + y > 0 Ex: x = -2, y = 3, x + y = -2 + 3 = 1 > 0 => TRUE	-Case 2: x = 0 I. x < y If x = 0, then x < y as y > 0 Ex: x = 0, y = 2 => 0 < 2 => TRUE II. x < \|y\| If x = 0, then \|y\| will be > x as \|y\|> 0 and x = 0 Ex: x = 0, y = 2 => 0 < \|2\| => TRUE III. x + y > 0 Now, x = 0 and y > 0 => x + y > 0 Ex: x = 0, y = 2, x + y = 0 + 2 = 2 > 0 => TRUE	-Case 3: x > 0 I. x < y If x > 0, then x < y as y > \|x\| => y > x Ex: x = 2, y = 3 => 2 < 3 => TRUE II. x < \|y\| If x > 0, then \|y\| will be > x as y > \|x\| Ex: x = 2, y = 3 => 2 < \|3\| => TRUE III. x + y > 0 Now, x > 0 and y > 0 => x + y > 0 Ex: x = 2, y = 3, x + y = 2 + 3 = 5 > 0 => TRUE

If |x| < y, then which of the following must be true?

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If |x| < y, then which of the following must be true?