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1. The drama club is selling tickets to their play to raise money for the show’s expenses. One solution is the place where the two lines cross, which reflects the minimal number of student and adult tickets needed to achieve the minimum ticket revenue criterion. This point resolves the inequality system.

   What is the solution to a system of inequalities?

   Generally, To write the system of inequalities, we need to represent the two conditions that must be satisfied: the total number of tickets sold must be less than or equal to 140 and the total ticket revenue must be at least $1100.

   First, let’s consider the total number of tickets sold. Since each student ticket sells for $6.50 and each adult ticket sells for $10, the total number of student tickets sold is 6.50xx and the total number of adult tickets sold is 10yy. Therefore, the total number of tickets sold is 6.50xx + 10yy. We can express this as an inequality as follows:

   6.50xx + 10yy ≤ 140

   Next, let’s consider the total ticket revenue. Since each student ticket sells for $6.50 and each adult ticket sells for $10, the total revenue from student ticket sales is 6.50xx and the total revenue from adult ticket sales is 10yy. Therefore, the total ticket revenue is 6.50xx + 10yy. We can express this as an inequality as follows:

   6.50xx + 10yy ≥ 1100

   Now we have a system of two inequalities:

   6.50xx + 10yy ≤ 140 6.50xx + 10yy ≥ 1100

   To solve this system of inequalities graphically, we can plot the two inequalities on the same coordinate plane and find the region of the plane that satisfies both inequalities.

   For the first inequality, 6.50xx + 10yy ≤ 140, we can solve for yy in terms of xx by dividing both sides of the inequality by 10:

   yy ≤ (140 – 6.50xx)/10

   This is the equation for a line with slope -6.5 and y-intercept 14. To graph this inequality, we can plot the line and shade the region above the line.

   For the second inequality, 6.50xx + 10yy ≥ 1100, we can solve for yy in terms of xx by dividing both sides of the inequality by 10:

   yy ≥ (1100 – 6.50xx)/10

   This is the equation for a line with slope -6.5 and y-intercept 110. To graph this inequality, we can plot the line and shade the region below the line.

   The region of the plane that satisfies both inequalities is the intersection of the two shaded regions. This is the area where the number of student tickets and the number of adult tickets can be varied to satisfy both conditions.

   One possible solution is the point where the two lines intersect, which represents the minimum number of student tickets and adult tickets that must be sold to meet the minimum ticket revenue requirement. This point will be the solution to the system of inequalities.

   Read more about solving a system of inequalities

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