#### Answer

$x=\dfrac{3}{2}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given equation, $
|x-9|=|x+6|
,$ use the definition of an absolute equality. Then use the properties of equality to isolate the variable.
$\bf{\text{Solution Details:}}$
Since $|x|=|y|$ implies $x=y \text{ or } x=-y,$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
x-9=x+6
\\\\\text{OR}\\\\
x-9=-(x+6)
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
x-9=x+6
\\\\
x-x=6+9
\\\\
0=15
\text{ (FALSE)}
\\\\\text{OR}\\\\
x-9=-(x+6)
\\\\
x-9=-x-6
\\\\
x+x=-6+9
\\\\
2x=3
\\\\
x=\dfrac{3}{2}
.\end{array}
Hence, $
x=\dfrac{3}{2}
.$