WholeLottaLove wrote:
Hello, I am a bit confused regarding absolute value.
If |x|=|2y|, then why why aren't x and 2y both positive? If the abs. value of something (i.e. it's positive value) is equal to something else, doesn't that imply that they are both positive? For example, if x=|2y| doesn't that mean that x is positive?
Also, for #2, xy both have the same signs. If x and y are negative, why would |x|=|2y| become -x = -2y? I get that it's equal to |x|=|2y| but why even take that step? |-x| = |-2y| will always be positive, right?
If |x|=|2y|, then why why aren't x and 2y both positive? If the abs. value of something (i.e. it's positive value) is equal to something else, doesn't that imply that they are both positive? For example, if x=|2y| doesn't that mean that x is positive?
Also, for #2, xy both have the same signs. If x and y are negative, why would |x|=|2y| become -x = -2y? I get that it's equal to |x|=|2y| but why even take that step? |-x| = |-2y| will always be positive, right?
Bunuel wrote:
|x|=|2y|, what is the value of x-2y?
First of all |x|=|2y| means that either x=2y or x=-2y.
(1) x+2y = 6. Now, the second case is not possible since if x=-2y then from this statement we would have that -2y+2y=6 --> 0=6, which obviously is not true. So, we have that x=2y, in this case x-2y=2y-2y=0. Sufficient.
(2) xy>0 --> x and y are either both positive or both negative, in any case |x|=|2y| becomes x=2y (if x and y are both negative then |x|=|2y| becomes -x=-2y which is the same as x=2y). Now, if x=2y then x-2y=2y-2y=0. Sufficient.
Answer: D.
Hope it's clear.
First of all |x|=|2y| means that either x=2y or x=-2y.
(1) x+2y = 6. Now, the second case is not possible since if x=-2y then from this statement we would have that -2y+2y=6 --> 0=6, which obviously is not true. So, we have that x=2y, in this case x-2y=2y-2y=0. Sufficient.
(2) xy>0 --> x and y are either both positive or both negative, in any case |x|=|2y| becomes x=2y (if x and y are both negative then |x|=|2y| becomes -x=-2y which is the same as x=2y). Now, if x=2y then x-2y=2y-2y=0. Sufficient.
Answer: D.
Hope it's clear.
The absolute value cannot be negative |some \ expression|\geq{0}, or |x|\geq{0} (absolute value of x, |x|, is the distance between point x on a number line and zero, and the distance cannot be negative).
So, if given that x=|2y| then x must be more than or equal to zero (RHS is non-negative thus LHS must also be non-negative).
But in our case we have that |x|=|2y|. In this case x and/or y could be negative. For, example x=-2 and y=-1 --> |x|=2=|2y|.
As for (2):
When x\leq{0} then |x|=-x, or more generally when some \ expression\leq{0} then |some \ expression|\leq{-(some \ expression)}. For example: |-5|=5=-(-5);
When x\geq{0} then |x|=x, or more generally when some \ expression\geq{0} then |some \ expression|\leq{some \ expression}. For example: |5|=5.
So, if x<0 and y<0, then |x|=-x and |2y|=-2y --> -x=-2y --> x=2y. If x>0 and y>0, then |x|=x and |2y|=2y --> x=2y, the same as in the first case.
For more check Absolute Value chapter of Math Book: math-absolute-value-modulus-86462.html
DS questions on absolute value to practice: search.php?search_id=tag&tag_id=37
PS questions on absolute value to practice: search.php?search_id=tag&tag_id=58
Tough absolute value and inequity questions with detailed solutions: inequality-and-absolute-value-questions-from-my-collection-86939.html
Hope it helps.