Question

Diego lives in a city and Anya lives in another city. Their houses are 84 miles apart. They both meet at their favorite restaurant, which is 4x-4 miles from Diego’s house and 3x+11 miles from Anya’s house. Diego argues that in a straight line distance, the restaurant is exactly halfway between his house and Anya’s house. Is Diego right? Justify your reasoning.

(a) Yes, the restaurant is exactly halfway.
(b) No, the restaurant is closer to Diego's house.
(c) No, the restaurant is closer to Anya's house.
(d) It depends on the values of x.

Answer 1

(d) It depends on the values of x, as the distances from Diego's and Anya's houses to the restaurant, given by 4x-4 and 3x+11 respectively, must total 84 miles when combined. Solving for x reveals that the restaurant is not exactly halfway between their houses.

Step-by-step explanation:

The correct answer is option (d) It depends on the values of x. To determine if the restaurant is exactly halfway between Diego’s and Anya’s houses, we need to set the distances from their respective houses to the restaurant equal to each other, that is 4x - 4 miles for Diego and 3x + 11 miles for Anya. The sum of these two distances should be equal to the total distance between their houses, which is given as 84 miles:

• For Diego: 4x - 4 miles
• For Anya: 3x + 11 miles

If we add these two expressions, we get 4x - 4 + 3x + 11 = 84. Simplifying this equation we get 7x + 7 = 84, or 7x = 77. Dividing both sides of the equation by 7, we find x = 11.

Plugging the value of x back into the distances, we get 4(11) - 4 = 40 miles for Diego and 3(11) + 11 = 44 miles for Anya. Evidently, the distances are not equal, so the restaurant is not exactly halfway between their houses. Therefore, the position of the restaurant in relation to their houses indeed depends on the value of x.