<s>[INST] 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?
 [/INST] Plug $x=11$ into the original equation and square both sides. Do the same with $x=5$. We obtain
$$(11+a)^2=b^2=(5+a)^2.\tag1$$
Expanding (1) and solving for $a$ we get
$$121+22a+a^2=25+10a+a^2\ \Longleftrightarrow\ a=-8$$
Plugging this value of $a$ back into the original equation, this gives
$$b=\vert 11+a\vert=\vert 5+a\vert=3.$$
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