Solve for \(x\) in the equation \(3(x+2)-8=(5x+4)-2(x+1)\text{.}\)

To solve for \(x\text{,}\) we will first need to simplify the left side and right side of the equation as much as possible by distributing and combining like terms:

\begin{align*} 3(x+2)-8\amp=(5x+4)-2(x+1)\\ 3x+6-8\amp=5x+4-2x-2\\ 3x-2\amp=3x+2 \end{align*}

From here, we'll want to subtract \(3x\) from each side:

\begin{align*} 3x-2\subtractright{3x}\amp=3x+2\subtractright{3x}\\ -2\amp=2 \end{align*}

As the equation \(-2=2\) is not true for any value of \(x\text{,}\) there is no solution to this equation. We write the solution set as: \(\emptyset\text{.}\)