# How do you solve abs(6-x)=9?

Sep 29, 2015

Remember the rule:
$| x | = a$
If $x < 0$
${a}_{1} = - x$
If $x > 0$
${a}_{2} = x$
So if $x$'s interval isn't determined we will have 2 results.

#### Explanation:

$6 - x > 0$

$| 6 - x | = 6 - x$
$6 - x = 9$
${x}_{1} = - 3$

$6 - x < 0$

$| 6 - x | = x - 6$
$x - 6 = 9$
${x}_{2} = 15$

Sep 29, 2015

$x = 15$ and $x = - 3$.

#### Explanation:

Absolute value equations have two solutions; since absolute values will always be positive, this makes sense. For example, in $\left\mid x \right\mid = 9$, $x$ can equal 9 and -9; the absolute value of 9 equals 9 and the absolute value of -9 equals 9.

As such, we need two equations to find the two solutions. Our two equations are: $6 - x = 9$ and $6 - x = - 9$. You've probably noticed that these equations are extremely similar - except one equation equals 9, and the other -9. Always set up absolute value equations like this when you're ready to solve.

Let's get to the solving, starting with $6 - x = 9$:

$6 - x = 9$ (original equation)
$- x = 3$ (subtracting 6 from both sides)
$x = - 3$ (dividing by -1)

Alright, $x = - 3$ is one solution. Now, for $6 - x = - 9$:

$6 - x = - 9$ (original equation)
$- x = - 15$ (subtracting 6 from both sides)
$x = 15$ (dividing by -1)

And that's it. Our solutions are $x = 15$ and $x = - 3$.