answer:Start by multiplying both sides of the equation by 7x to eliminate the denominators, giving: [ x^2 + 2 = 7x. ] Rearranging the terms gives: [ x^2 - 7x + 2 = 0 ] To solve this quadratic equation, use the quadratic formula: [ x = frac{-b pm sqrt{b^2 - 4ac}}{2a} ] Here, a = 1, b = -7, and c = 2. Plugging in these values gives: [ x = frac{-(-7) pm sqrt{(-7)^2 - 4 times 1 times 2}}{2 times 1} ] [ x = frac{7 pm sqrt{49 - 8}}{2} ] [ x = frac{7 pm sqrt{41}}{2} ] The solutions are: [ x = frac{7 + sqrt{41}}{2} text{ and } x = frac{7 - sqrt{41}}{2} ] Since sqrt{41} is approximately 6.4, the positive value frac{7 - sqrt{41}}{2} is approximately 0.3, which is the smallest positive value. The final smallest value of x, using the exact form is boxed{frac{7 - sqrt{41}}{2}}.

answer:Given that the fraction frac{7}{x-1} must be an integer, we start by considering the divisors of 7. Since 7 is a prime number, its only positive divisors are 1 and 7. However, we must also consider the negative divisors, which are -1 and -7. Therefore, we have: 1. For the fraction to be an integer, x-1 must equal one of the divisors of 7. This gives us the equations: - x - 1 = 1 - x - 1 = 7 - x - 1 = -1 - x - 1 = -7 2. Solving each equation for x gives us the possible values of x: - From x - 1 = 1, we get x = 2. - From x - 1 = 7, we get x = 8. - From x - 1 = -1, we get x = 0. - From x - 1 = -7, we get x = -6. 3. The sum of all integers x that satisfy the condition is 0 + 2 - 6 + 8. 4. Calculating the sum gives us 0 + 2 - 6 + 8 = 4. Therefore, the sum of all integers x that make the fraction frac{7}{x-1} an integer is boxed{4}, which corresponds to choice B.

answer:First, let's calculate the time it would take for the Pinedale bus line to travel 25 kilometers at an average speed of 60 km/h. Time = Distance / Speed Time = 25 km / 60 km/h To convert the speed into km/min, since we are given the stops in minutes, we divide 60 km/h by 60 (since there are 60 minutes in an hour): 60 km/h ÷ 60 = 1 km/min Now we can calculate the time: Time = 25 km / 1 km/min Time = 25 minutes Since the bus has stops every 5 minutes, we can find out how many stops are made in 25 minutes by dividing the total time by the time between stops: Number of stops = Total time / Time between stops Number of stops = 25 minutes / 5 minutes/stop Number of stops = 5 stops Therefore, the Pinedale mall is boxed{5} stops away from Yahya's house.

answer:To solve this problem, let's analyze the speed and alignment of the clock hands. 1. We know a Martian day is divided into 13 hours. 2. In 13 hours, the hour hand makes one complete revolution, i.e., it moves 360 degrees. 3. The minute hand makes 13 complete revolutions in 13 hours, as it completes a full rotation each hour. Thus, it moves (13 times 360) degrees in a Martian day. We need to determine how often the hands align perfectly. Step 1: Hour and Minute Hands 1. The minute hand aligns with the hour hand 12 times in 13 hours, because: [ 13 - 1 = 12 ] This is because the minute hand needs to lap the hour hand once every hour (12 laps for 12 hours). Step 2: Inclusion of the Second Hand 1. After the Martian Rabbit altered the second hand's speed in the tea, consider it aligns a whole number (n) times with the hour hand in 13 hours. 2. Each time all three hands align: - Let (d = text{GCD}(12, n)). - This means every ( frac{text{total time}}{d}): Given the problem constraints, these alignments occur simultaneously (d) times in a Martian day. Calculation of Possible Alignments To determine the value of (d) and thus the number of times all three hands align: - The possible values for (d) are the divisors of 12. **Divisors of 12**: [ 12 = 2^2 times 3 implies text{Divisors: } 1, 2, 3, 4, 6, 12 ] # Conclusion Hence, the possible number of times all three hands can align in one Martian day are (6) possible solutions, which are the divisors of 12. [ boxed{1, 2, 3, 4, 6, 12} ]