Question

1. Suppose U = {1, 2, 3, 4, 5, 6, 7, 8} is the universal set and P = {1, 2, 3, 4}. What is P'?

2. Solve the inequality: 4x- 3 ≥ 21.
3. Your goal is to save at least $360.00 over the next 6 weeks. How much money must you save each week to meet that goal? Write and solve an inequality.
4. Of 256 consumers polled, some like only thriller movies, some prefer just comedies, and some like both. If 78 people like only thrillers, and 116 like both, how many people prefer only comedies?
5. Solve the absolute value equation: |x − 10| = 5

Answer 1

[AI-generated answer — Bot GPT 3.5]

1. P' is the complement of P, which consists of all the elements in the universal set U that are not in P. Therefore, P' = {5, 6, 7, 8}.

2. To solve the inequality 4x - 3 ≥ 21, first add 3 to both sides to isolate the variable: 4x ≥ 24. Then, divide both sides by 4 to solve for x: x ≥ 6.

3. Let x represent the amount of money to save each week. Since the goal is to save at least $360.00 over 6 weeks, the inequality would be 6x ≥ 360. Divide both sides by 6 to solve for x: x ≥ 60. Therefore, you must save at least $60.00 each week to meet the goal.

4. Let T represent the number of people who like only thriller movies, C represent the number of people who like only comedies, and B represent the number of people who like both. Since the total number of people polled is 256, we can set up the following equation: T + C + B = 256.
Given that 78 people like only thrillers (T = 78) and 116 like both (B = 116), we can substitute these values into the equation: 78 + C + 116 = 256.
Simplifying, we have C + 194 = 256. Subtracting 194 from both sides, we get C = 62. Therefore, 62 people prefer only comedies.

5. To solve the absolute value equation |x - 10| = 5, we consider two cases:
Case 1: x - 10 = 5. In this case, we can add 10 to both sides to solve for x: x = 15.
Case 2: -(x - 10) = 5. Here, we need to distribute the negative sign: -x + 10 = 5. Subtracting 10 from both sides, we find -x = -5. Finally, multiplying both sides by -1, we get x = 5.
Therefore, the solutions to the absolute value equation are x = 15 and x = 5.