3) 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:
i. You want to stay at least 5 nights
ii. At least one night will be spent at a resort
iii. Resorts average $200 per night and regular stay hotels average $100 per night. Your budget does not exceed $700 for lodging.
b. Graph the solution set of your system of inequalities
c. Based on your graph, how many nights can you spend at a resort and stay within your budget?

1. x + y > 5

2. x > 1

3. 200x + 100y < 700

Cannot graph here.

To write a system of inequalities that models the given conditions, we need to consider the following:

i. You want to stay at least 5 nights:
This condition can be represented by the inequality:
x + y ≥ 5

ii. At least one night will be spent at a resort:
This condition can be represented by the inequality:
x ≥ 1

iii. The budget for lodging is not to exceed $700:
This condition can be represented by the inequality:
200x + 100y ≤ 700

Thus, the system of inequalities is:

x + y ≥ 5
x ≥ 1
200x + 100y ≤ 700

To graph the solution set of these inequalities, we can plot the lines corresponding to each inequality on a coordinate plane and shade the region that satisfies all the conditions.

For the first inequality, x + y ≥ 5, we can rewrite it in slope-intercept form:
y ≥ -x + 5

To graph it, plot the line y = -x + 5 and shade the region above the line.

For the second inequality, x ≥ 1, the line will be vertical and pass through x = 1. Shade the region to the right of the line.

For the third inequality, 200x + 100y ≤ 700, rewrite it in slope-intercept form:
y ≤ -2x + 7

Graph the line y = -2x + 7 and shade the region below the line.

The solution set of the system of inequalities is the region where all three shaded regions overlap.

To determine the number of nights you can spend at a resort and stay within your budget, look at the graph and find the point(s) within the overlapping region that satisfy the condition of staying within the budget. The x-coordinate of the point(s) will represent the number of nights you can spend at a resort.