Answer: \(x=-\dfrac13\) or \(x=15\).

The key point is to split the absolute-value expression into the correct cases before solving.

Since the two absolute values are equal,

\(4x+9=6-5x\)

or

\(4x+9=-(6-5x).\)

From the first equation,

\(9x=-3,\)

so

\(x=-\frac13.\)

From the second equation,

\(4x+9=-6+5x,\)

so

\(x=15.\)