Absolute Value (with variables)

The absolute value of a number is always positive.  It represents the number's distance from 0, and distance is always positive.

However, when you are working with variables and absolute value, you have to remember that while the absolute value of a number is always positive, the original number may have been negative.

Think about this:

$\mid 6 \mid = 6$

So, if you are given $\mid x \mid = 6$, then $x$ might equal $6$.

But, remeber that:

$\mid ^-6 \mid = 6$

So, if you are given $\mid x \mid = 6$, then $x$ might equal $6$, or $x$ might equal $^-6$.

And, that's the most important thing to remember when working with variables and absolute value, there are always two possible values for the variable, one is negative and one is positive.

So:

$$\eqalign{\mid x \mid &= 6\\x&=6 \text{ or } ^-6}$$

The possibility of two answers is always true for an absolute value that contains a variable. 

$$\eqalign{\mid x \mid &= 532\\x&=532 \text{ or } ^-532}$$

$$\eqalign{\mid x \mid &= .78\\x&=.78 \text{ or } ^-.78}$$

$$\eqalign{\mid x \mid &= \dfrac{5}{7}\\x&=\dfrac{5}{7} \text{ or } ^-\dfrac{5}{7}}$$

$$\eqalign{\mid x \mid &= y\\x&=y \text{ or }^ -y}$$

No matter what kind of number an absolute value equals, the variable could be positive or negative.