This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to the problems: 1. Inequalities i) Solve 4(2x+3) < 7-2x. Step 1: Distribute the 4 on the left side. 8x + 12 < 7 - 2x Step 2: Add 2x to both sides. 10x + 12 < 7 Step 3: Subtract 12 from both sides. 10x < 7 - 12 10x < -5 Step 4: Divide by 10. x < -(5)/(10) x < -(1)/(2) The solution is x < -(1)/(2). ii) Solve 3x+2 > 2x+5. Step 1: Subtract 2x from both sides. 3x - 2x + 2 > 5 x + 2 > 5 Step 2: Subtract 2 from both sides. x > 5 - 2 x > 3 The solution is x > 3. 2. Venn Diagram (Set Theory) Let G be the set of people who watched GTV, K for KTV, and M for MTV. Given: Total people = 70 |G| = 32 |K| = 24 |M| = 22 |G K| = 14 |K M| = 8 |G M| = 13 |G K M| = 5 a) Illustrate the information in a Venn diagram. First, calculate the "only" regions: |G K only| = |G K| - |G K M| = 14 - 5 = 9 |K M only| = |K M| - |G K M| = 8 - 5 = 3 |G M only| = |G M| - |G K M| = 13 - 5 = 8 Next, calculate the number of people who watched only one channel: |G only| = |G| - (|G K only| + |G M only| + |G K M|) = 32 - (9 + 8 + 5) = 32 - 22 = 10 |K only| = |K| - (|G K only| + |K M only| + |G K M|) = 24 - (9 + 3 + 5) = 24 - 17 = 7 |M only| = |M| - (|K M only| + |G M only| + |G K M|) = 22 - (3 + 8 + 5) = 22 - 16 = 6 The Venn diagram would show three overlapping circles for G, K, and M. • The center intersection (G K M) would be 5. • The intersection of G and K only would be 9. • The intersection of K and M only would be 3. • The intersection of G and M only would be 8. • The G only region would be 10. • The K only region would be 7. • The M only region would be 6. b) Find the number of people who: (i) did not watch TV at all Step 1: Calculate the total number of people who watched at least one channel. |G K M| = |G only| + |K only| + |M only| + |G K only| + |K M only| + |G M only| + |G K M| |G K M| = 10 + 7 + 6 + 9 + 3 + 8 + 5 = 48 Step 2: Subtract this from the total number of people interviewed. Did not watch TV = Total people - |G K M| Did not watch TV = 70 - 48 = 22 The number of people who did not watch TV at all is 22. (ii) watched MTV only From the calculations above, the number of people who watched MTV only is 6. 3. Ratio and Proportion Lois and Gina share 100.00 in the ratio 2:5. Step 1: Find the total number of parts in the ratio. Total parts = 2 + 5 = 7 Step 2: Calculate Lois's share. Lois's share = (2)/(7) × \100.00 = (\200.00)/(7) ≈ \28.57 Step 3: Calculate Gina's share. Gina's share = (5)/(7) × \100.00 = (\500.00)/(7) ≈ \71.43 Lois receives \28.57 and Gina receives \71.43. 4. Table Completion (Function Evaluation) Complete the table for the relation y = 5 + 3x - x^2 from x = -2 to x = 5. | x | y = 5 + 3x - x^2 | |-----|--------------------| | -2 | 5 + 3(-2) - (-2)^2 = 5 - 6 - 4 = -5 | | -1 | 5 + 3(-1) - (-1)^2 = 5 - 3 - 1 = 1 | | 0 | 5 + 3(0) - (0)^2 = 5 + 0 - 0 = 5 | | 1 | 5 + 3(1) - (1)^2 = 5 + 3 - 1 = 7 | | 2 | 5 + 3(2) - (2)^2 = 5 + 6 - 4 = 7 | | 3 | 5 + 3(3) - (3)^2 = 5 + 9 - 9 = 5 | | 4 | 5 + 3(4) - (4)^2 = 5 + 12 - 16 = 1 | | 5 | 5 + 3(5) - (5)^2 = 5 + 15 - 25 = -5$ | The completed table is: |c|c|c|c|c|c|c|c|c| x & -2 & -1 & 0 & 1 & 2 & 3 & 4 & 5 \\ y & -5 & 1 & 5 & 7 & 7 & 5 & 1 & -5 \\ Last free one today — make it count tomorrow,