While traveling for vacation, you have decided to divide your lodging between a resort and a regular stay hotel. Let x represent the number of nights you will spend in a resort and let y represent the number of nights you will spend in a regular stay hotel.
A) Write a system of inequalities that models the following conditions:
1) You want to stay at least 5 nights
2) At least one night will be spent at a resort
3) Resorts average $200 per night and regular stay hotels average $100 per night. Your budget does not exceed $700 for lodging
B) How do I graph the solution set of my system of inequalities?
C) How many nights can you spend at resort and stay within your budget?
Help please!!!!
A) To write a system of inequalities that models the given conditions, we will break down the conditions into separate inequalities:
1) You want to stay at least 5 nights
This can be represented as:
x + y ≥ 5
2) At least one night will be spent at a resort
This can be represented as:
x ≥ 1
3) Your budget does not exceed $700 for lodging
Since resorts average $200 per night and regular stay hotels average $100 per night, the total cost of lodging can be calculated as:
Total cost = (200 * x) + (100 * y)
So, the budget constraint can be represented as:
(200 * x) + (100 * y) ≤ 700
Therefore, the system of inequalities that models the given conditions is:
x + y ≥ 5
x ≥ 1
(200 * x) + (100 * y) ≤ 700
B) To graph the solution set of the system of inequalities, we can plot the inequalities on a coordinate plane.
1) Start by graphing the line x + y = 5:
To do this, plot the points (0, 5) and (5, 0) and draw a line passing through these points.
2) The inequality x ≥ 1 represents a vertical line passing through the point (1, 0). We can draw this line as well.
3) For the inequality (200 * x) + (100 * y) ≤ 700, we can rewrite it as:
2x + y ≤ 7
This inequality can be plotted as a straight line with a y-intercept of 7 and a slope of -2.
Once all the lines have been graphed, the solution set will be the region that satisfies all three inequalities. Shade the region where all the lines intersect or overlap, and that will represent the solution set.
C) To determine how many nights you can spend at a resort and stay within your budget, you need to find the points within the solution set where the cost of lodging does not exceed $700.
Substitute different values of x and y into the inequality (200 * x) + (100 * y) ≤ 700 and calculate the corresponding cost. This will help you determine the maximum number of nights you can spend at a resort while staying within your budget.
For example, if x = 2 and y = 3, the total cost of lodging would be (200 * 2) + (100 * 3) = 700. Therefore, you can spend a maximum of 2 nights at a resort and 3 nights at a regular stay hotel while staying within your budget. Repeat this process for different values of x and y to explore other possibilities within your budget constraint.