1. The problem states that choosing a crime film and choosing a documentary film are mutually exclusive events. This means the two sets (Crime and Documentary) do not overlap. 2. In the Venn diagram, the overlap between Crime and Documentary is represented by the expressions inside both circles. Here, the overlap expressions are $3x - 15$ and $8x$. 3. Since the events are mutually exclusive, the overlap must be zero. Therefore, we set the expressions equal to zero: $$3x - 15 = 0$$ and $$8x = 0$$ 4. Solve each equation: From $$3x - 15 = 0$$: $$3x = 15$$ $$x = 5$$ From $$8x = 0$$: $$x = 0$$ 5. The value of $x$ must satisfy both overlap expressions being zero. Since $8x=0$ implies $x=0$, and $3x-15=0$ implies $x=5$, these contradict each other. 6. This suggests a possible error in the problem statement or diagram because the overlap cannot be two different expressions simultaneously if the events are mutually exclusive. 7. However, if we consider the overlap as a single region, it must be zero. Let's assume the overlap is represented by the sum of the two expressions $3x - 15$ and $8x$ and set their sum to zero: $$ (3x - 15) + 8x = 0 $$ Simplify: $$ 11x - 15 = 0 $$ $$ 11x = 15 $$ $$ x = \frac{15}{11} $$ 8. Therefore, the value of $x$ that makes the overlap zero (mutually exclusive) is $\frac{15}{11}$. Final answer: $x = \frac{15}{11}$