Originally Posted by
Lambin Sure. When you think about absolute value functions, by definition, we know that the outcome is always positive no matter whether we have a negative or positive value on the inside.
For example,
|3| = 3 and |-3| = 3.
It didn't matter whether it was a 3 or a -3, the outcome is a positive 3.
So, if we turned things around and wanted to solve for an unknown variable in an absolute value function:
|x| = ?
How would we know whether the x was a positive or negative value? Regardless of whether it is positive or negative, the outcome is going to be the same—it will be a positive outcome. So because they share the same outcome, we consider both the positive and negative value as solutions of the absolute-value function that are both valid. In this latter example, it would mean that we have to consider BOTH x and -x as solutions.
How does this connect with our problem in this post?
When we work with analytic expressions instead of numbers in absolute-value functions, we don't treat them differently.
So, with $\displaystyle |4x+3|=y$, we have to consider two things:
1. $\displaystyle 4x+3=y$
2. $\displaystyle -(4x+3)=y$
See?
This would mean that for $\displaystyle |3x-2y+10|+|2x-3y|=10$, you would discover that there would be four equations. Think about what was shown earlier, and see how it relates with this problem. With these four equations, we have a system of equations with $\displaystyle x^3(1-y)+y^3(1-x)=12xy+18$.
You would have to solve from here—seeing where the four equations intersect with $\displaystyle x^3(1-y)+y^3(1-x)=12xy+18$.