<s>[INST] How do you solve two equal absolute value expressions? I'm having trouble understanding how the following is solved. 
$$|x+1| = |x-2|$$
 [/INST] So we know what absolute value means. It means that we want the positive version of $x+1$ and $x-2$.
So we don't know what the signs of $x-2$ and $x+2$ are.
Thus, we can make four different equalities, of which $0$, $1$, $2$, $3$, or $4$ may be true. There may be multiple values of $x$ or there might not be any value of $x$ that satisfies.
First, $x+1 = x-2$ simply implies that $x+3 = x$ which is not possible.
Secondly, $-(x+1) = x-2$ implies that $2x=1$, so $x = 1/2$.
Thirdly, $(x+1)=-(x-2)$. This case is the same as the second case because we can multiply both sides by $-1$.
Fourthly, $-(x+1)=-(x-2)$ implies that $x+1=x-2$, which is the same as the first case.
Thus, the only possible answer is in cases 2 and 3 for which $x = 1/2$. To check, note that $1+1/2 = 3/2$ and $1/2-2= -3/2$. $3/2$ and $-3/2$ have the same absolute value.
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