# How can an absolute value equation with a variable have both a positive, and negative answer?

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?

## 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.

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 $|-12| = -(-12) = 12.$

• Thank you so much! Yes I didn't notice when the question said the answer could have both 12, and -12 it was not using absolute value bars, and was just referring to how the variable can be both positive, and negative. This answer my question! – Christie Apr 11 '16 at 7:26
• I'm glad you got it. Good luck! – Em. Apr 11 '16 at 7:27

$7-y=12$ or $7-y=-12$

$y=-5$ or $y=19$

If $y=-5$ then $|7-(-5)|=|12|=12$ and if $y=19$, then $|7-19|=|-12|=12$

$|7-y|=12$ means that the distance (as you point out, this distance can only be nonnegative) between 7 and y is 12. Now, in $\mathbb{R}$ (the space of real numbers), there are two values of $y$ that are 12 units distant from 7. One of them is 12 + 7 = 19, while the other is -12 + 7 = -5.

Even though you have not asked for this in your OP, if we are talking about points in $\mathbb{R}^2$ and consider the "7" to be (7,0), the equation corresponds to the points on a circle centered at (7,0) with radius 12. In this case, there will be infinite number of solutions to this equation.

Even if the answer inside the vertical bars was negative, wouldn't the vertical bars make it positive, resulting in just 1 positive answer?

The vertical bars are there to express a property you want your $y$ to have. They don't influence what $y$ itself is.

A solution to the equation means the value of the $y$ you start your computation with, before subtracting it from $7$ and positivising the result. Both of those things happen during the calculation you want to work, but doing them doesn't change the fact that the $y$ you started out with might have been $-5$.