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If we get rid of the absolute values, we are left with the following 4 equations (using the logic that if $|a|=b$, then $a$ is either $b$ or $-b$): $$\begin{array}{r}3x+4y=12\\ -3x+4y=12\\ 3x-4y=12\\ -3x-4y=12\end{array}$$ We can then put these equations in slope-intercept form in order to graph them. $$\begin{array}{r}3x+4y=12⟹y=-\frac{3}{4}x+3\\ -3x+4y=12⟹y=\frac{3}{4}x+3\\ 3x-4y=12⟹y=\frac{3}{4}x-3\\ -3x-4y=12⟹y=-\frac{3}{4}x-3\end{array}$$ Now you can graph the lines to find the shape of the graph: We can easily see that it is a rhombus with diagonals of $6$ and $8$. The area is $\frac{1}{2}\times 6\times 8$, or $\boxed{24}$