answer:The inequality given is 2 leq |x-1| leq 5. We can interpret this as the absolute value of x-1 being at least 2 and at most 5. To solve this, we break it into two separate inequalities based on the definition of absolute value: 1. **Case 1: x-1 geq 0** - Here, |x-1| = x-1. Thus, the inequality 2 leq |x-1| leq 5 becomes: [ 2 leq x-1 leq 5 ] - Adding 1 to all parts of the inequality: [ 2+1 leq x-1+1 leq 5+1 ] [ 3 leq x leq 6 ] 2. **Case 2: x-1 < 0** - Here, |x-1| = -(x-1) = 1-x. Thus, the inequality 2 leq |x-1| leq 5 becomes: [ 2 leq 1-x leq 5 ] - Solving 2 leq 1-x: [ 2 leq 1-x implies x leq 1-2 implies x leq -1 ] - Solving 1-x leq 5: [ 1-x leq 5 implies x geq 1-5 implies x geq -4 ] - Combining these results: [ -4 leq x leq -1 ] Combining the results from both cases, the set of x-values satisfying the inequality is: [ 3 leq x leq 6 quad text{or} quad -4 leq x leq -1 ] Thus, the correct answer is boxed{textbf{(A)} -4leq xleq-1text{ or }3leq xleq 6}.

answer:1. **Understanding the problem**: Julia initially had paint for 45 rooms, but after losing four cans, she could cover only 36 rooms. We need to find out how many cans she used for these 36 rooms. 2. **Relating cans of paint to room coverage**: Losing four cans reduced her capacity by 9 rooms. Thus, each can originally covered frac{9}{4} rooms. 3. **Calculating the number of cans used for 36 rooms**: Since each can covers frac{9}{4} rooms, the number of cans required to paint 36 rooms is calculated by: [ text{Number of cans} = frac{36 text{ rooms}}{frac{9}{4} text{ rooms per can}} = 36 times frac{4}{9} = 16 ] 4. **Conclusion**: Julia used 16 cans of paint to cover 36 rooms. Thus, the final answer is boxed as 16. The final answer is boxed{C}.

answer:Given the quadratic equation x^{2}-2x-8=0, we can use the sum and product of roots formula for a quadratic equation ax^2+bx+c=0, where the sum of the roots x_1 + x_2 = -frac{b}{a} and the product of the roots x_1x_2 = frac{c}{a}. For our equation, a=1, b=-2, and c=-8. Therefore, we have: 1. The sum of the roots: x_1 + x_2 = -frac{-2}{1} = 2. 2. The product of the roots: x_1x_2 = frac{-8}{1} = -8. We are asked to find frac{x_1+x_2}{x_1x_2}. Substituting the values we found: [ frac{x_1+x_2}{x_1x_2} = frac{2}{-8} = -frac{1}{4}. ] Therefore, the answer is boxed{-frac{1}{4}}.

answer:According to the problem, let's assume the other factor is x+a. Then, (x-1)(x+a) = x^2 + (a-1)x - a = x^2 - 5x + c, therefore a-1 = -5, and c = -a, Solving these, we get a = -4, and c = 4. Therefore, the answer should be boxed{4}.