The inequality is $4\left|x-\frac34\right|+5\left|y+\frac65\right|\le10$, a rhombus centered at $\left(\frac34,-\frac65\right)$ whose vertices are $\frac{10}4$ away horizontally and $\frac{10}5$ away vertically. Its diagonals are $5$ and $4$, so the area is $\frac12\cdot5\cdot4=10$.
