I thought absolute values are supposed to be a number's distance from 0, which is always positive. So I had this equation:
| 7 – y | = 12
According to practice tests they say this,
This equation means we need to solve two equations: 7–y=12, and 7–y=-12 which means y=-5,19.
Even if the answer inside the vertical bars was negative, wouldn't the vertical bars make it positive, resulting in just 1 positive answer?
Thanks for any help you can give!
Edit
I found out what happened! I didn't notice when they said the equations could have both 12, and -12 they didn't include absolute value bars, so they were just referring to how the variable can be both positive, or negative.
Best Answer
Not quite.
Recall that $$|y| = \begin{cases}y,&\text{if } y\geq 0\\-y,&\text{if } y<0\end{cases}$$
Notice that the problem asks for all values $y$ such that $$|7-y| = 12.\tag 1$$ If $y = -5$, then $$|7-(-5)| = |7+5| = |12| = 12.$$ Further, if $y = 19$, then $$|7-(19)| = |-12| = -(-12) = 12$$ using the rule above.
Hence the values of $y$ that satisfy equation $(1)$ are $-5$ and $19$.
So although $y = 19$ gives $|-12|$, it still satisfies equation $(1)$ since we take the absolute value: since $-12<0$, we have that $|-12| = -(-12) = 12.$