Ok, so I get that Abs. value cannot be negative...distance cannot have a negative value.
We are trying to solve for x-2y, so naturally we are trying to determine x-2y. So,
If x=-2y (the absolute value of 2y) then the value of x-2y = -2y-2y = -4y, correct?
When x\leq{0} then |x|=-x. What you're saying is that, for example, |-4| = -(-4) or |-4| = 4. What is the point of writing |-4| = -(-4)
One final thing...In the stem you derived x=2y, x=-2y. Okay, but in #2. one of the cases is xy>0 so we could have -x and -y. If x and y are negative, doesn't that mean that you would substitute -x and y in to get -x=-2(-y) = -x=2y?
I'm sorry for being such a dolt. Sometimes, concepts that I know are very simple are extremely difficult to understand.
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?
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.
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.htmlDS questions on absolute value to practice:
search.php?search_id=tag&tag_id=37PS questions on absolute value to practice:
search.php?search_id=tag&tag_id=58Tough absolute value and inequity questions with detailed solutions:
inequality-and-absolute-value-questions-from-my-collection-86939.htmlHope it helps.