1. **Problem statement:** (a) A squash club has 27 members. 19 have black hair, 14 have brown eyes, and some have both black hair and brown eyes. (i) Place this information on a Venn diagram. (ii) Find the number of members with: I. Black hair or brown eyes. II. Black hair but not brown eyes. (b) Solve the inequality: $$2(3x + 2) - 20 > 8(x - 3)$$ --- 2. **Step (a)(i): Venn diagram setup** - Let $B$ be the set of members with black hair. - Let $E$ be the set of members with brown eyes. - Given: $|B| = 19$, $|E| = 14$, total members $= 27$. - Let $x$ be the number of members with both black hair and brown eyes, i.e., $|B \cap E| = x$. 3. **Step (a)(ii) calculations:** - Using the principle of inclusion-exclusion: $$|B \cup E| = |B| + |E| - |B \cap E| = 19 + 14 - x = 33 - x$$ - Since total members are 27, and all members are in $B$ or $E$ or neither, we have: $$|B \cup E| \leq 27$$ - So: $$33 - x \leq 27 \implies x \geq 6$$ - The problem states "and it have both black hair and brown eyes" but does not specify the exact number, so we take $x=6$ as the minimum consistent value. 4. **Number of members with black or brown eyes:** $$|B \cup E| = 33 - 6 = 27$$ So, all members have black hair or brown eyes. 5. **Number of members with black hair but not brown eyes:** $$|B \setminus E| = |B| - |B \cap E| = 19 - 6 = 13$$ 6. **Step (b): Solve the inequality** $$2(3x + 2) - 20 > 8(x - 3)$$ Expand both sides: $$6x + 4 - 20 > 8x - 24$$ Simplify: $$6x - 16 > 8x - 24$$ Bring all terms to one side: $$6x - 16 - 8x + 24 > 0$$ $$-2x + 8 > 0$$ Subtract 8: $$-2x > -8$$ Divide by -2 (remember to reverse inequality): $$x < 4$$ **Final answers:** - (a)(i) Venn diagram shows 6 members in both sets, 13 only black hair, 8 only brown eyes. - (a)(ii) I. Number with black or brown eyes: 27 - (a)(ii) II. Number with black hair but not brown eyes: 13 - (b) Solution to inequality: $x < 4$