Answer:To make a profit of at least $500 from selling tickets to the school dance, the ninth graders will need to earn a total of at least $500 from ticket sales. If they earn $3 for each ticket purchased in advance and $4 for each ticket purchased at the door, we can use the following equation to determine how many tickets they need to sell:
(Number of tickets sold in advance * $3) + (Number of tickets sold at the door * $4) ≥ $500
Since the ninth graders estimate that at most 300 students will attend the dance, we know that the total number of tickets sold in advance and at the door must be less than or equal to 300. Therefore, we can set up the following system of linear inequalities to represent the problem:
Number of tickets sold in advance + Number of tickets sold at the door ≤ 300
(Number of tickets sold in advance * $3) + (Number of tickets sold at the door * $4) ≥ $500
To solve this system of inequalities, we can graph the two inequalities on the same coordinate plane and find the intersection of the two lines. The intersection represents the solutions to the system of inequalities.
The first inequality can be represented by a line in the form "y ≤ mx + b," where m is the slope and b is the y-intercept. The slope of the line is 1 and the y-intercept is 300, so the equation for the line is "y ≤ x + 300."
The second inequality can be represented by a line in the form "y ≥ mx + b," where m is the slope and b is the y-intercept. The slope of the line is -4/3 and the y-intercept is 500, so the equation for the line is "y ≥ -4/3x + 500."
When we graph these two lines on the same coordinate plane, we can see that the intersection of the lines represents the solutions to the system of inequalities. The intersection is the point where the two lines cross, and this point lies within the region defined by the inequalities.
This is all the information you need
Explanation: