Ok, so here is my question:
Let's say we look for all the cases of x,y being positive and negative (four in total)
(x is negative)
If x is negative, why don't we plug in -x for all values of x in both equations (i.e. x+|x|+y=7 ==> -x+-x+y=7)
Think about this:
Is 2x + 7 = 0 same as -2x + 7 = 0?
You know that these two are different equations and yield different values of x. x can be negative/positive.
On the other hand, how do you solve something like this: 2|x| + 7 = 0
You need the value of x, not of |x|. How will you get the value of x?
You will use the definition of |x|
|x| = x if x >= 0
|x| = -x if x < 0
So you take 2 cases:
x >= 0 so |x| = x
2|x| + 7 = 0 => 2x + 7 = 0
x = -7/2 (this doesn't work since x must not be negative)
x < 0 so |x| = -x
2|x| + 7 = 0 => 2(-x) + 7 = 0
x = 7/2 (this doesn't work either since x must be negative)
So there is no value of x that satisfies 2|x| + 7 = 0
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