build absolute value equations know solution We have absolute value equations with unknown coefficients:
$$|x + a| = b$$
and we know the solutions:
$$x = 11 \text{ and } x = 5$$
We need to find $a$ and $b$. From
$$11 + a = b \\
5 + a = -b$$
we get $a = -8$ and $b = 3$.
But we can try another way:
$$11 + a = -b \\
5 + a = b$$
and get $a =8$ and $b = -3$, which is not correct, apparently we need take absolute value from $b$.
How can I formulate this rule and explain it to another person?

We want to solve $|x + a| = b$ where we know that the solutions for $x$ are 11 and 5. Since it's an absolute value, $b$ must be positive and therefore $b > -b$. We might try to solve
$$11 + a = -b \qquad 5 + a = b$$
But $11 + a > 5 + a$ and $-b$ cannot be greater than $b$. So we are left with
$$11 + a = b \qquad 5 + a = -b$$
which can be solved using simultaneous equations to get $a = -8$ and $b = 3$.
