2008 AMC 12A Problems/Problem 14

Problem

What is the area of the region defined by the inequality $|3x-18|+|2y+7|\le3$ ?

Area is invariant under translation, so after translating left $6$ and up $3/2$ units, we have the inequality

which forms a diamond centered at the origin and vertices at $(\pm 1, 0), (0, \pm 1.5)$ . Thus the diagonals are of length $2$ and $3$ . Using the formula $A = \frac 12 d_1 d_2$ , the answer is $\frac{1}{2}(2)(3) = 3 \Rightarrow \mathrm{(A)}$ .