Tickets for the Spring Dance cost $3 per person or $5 per couple. To cover expenses, at least $750 worth of tickets must be sold. However, no more than 400 people can fit in the gym where the dance is being held. How do I solve this inequality?
To solve this inequality, we need to consider two factors: the number of people and the amount of money collected. Let's break it down step by step:
Step 1: Define the variables
Let's define two variables:
- x as the number of individuals attending the dance
- y as the number of couples attending the dance
Step 2: Set up the inequalities
Based on the given information, we can set up the following inequalities:
- Individual Tickets: Since each person pays $3, the total amount collected from individuals can be represented by 3x. This must be greater than or equal to $750, so we have the inequality: 3x ≥ 750.
- Couple Tickets: Since each couple pays $5, the total amount collected from couples can be represented by 5y. This must also be greater than or equal to $750, so we have the inequality: 5y ≥ 750.
- Capacity Limit: The total number of people attending the dance can be represented as x + 2y (because each couple consists of two people). This must not exceed the gym's capacity of 400, so we have the inequality: x + 2y ≤ 400.
Step 3: Simplify the inequalities
We don't need both inequalities involving money, so let's use the individual ticket inequality (3x ≥ 750) and simplify it by dividing both sides by 3: x ≥ 250.
Step 4: Graph the inequalities
To graph the system of inequalities, start by graphing the individual ticket inequality (x ≥ 250) as a vertical line passing through the point (250,0). Shade the area to the right of the line to represent all the possible values of x that satisfy the inequality.
Next, graph the couple ticket inequality (x + 2y ≤ 400) as a linear equation. First, rewrite it in slope-intercept form: 2y ≤ -x + 400. Then graph the line with a slope of -1/2, passing through the point (0, 200). Shade the area below the line to represent all the possible values of (x, y) that satisfy the inequality.
Step 5: Find the feasible region
The feasible region is the overlapping shaded region between the two inequalities. It represents the values that simultaneously satisfy both inequalities.
Step 6: Analyze the feasible region
Look for the intersection point(s) between the lines on the graph. This will give you the specific values for x and y that satisfy both inequalities. From this, you can determine the number of individuals (x) and couples (y) that should attend the dance to meet the requirements.
Note: Depending on the exact equations, the feasible region might be a single point or a shaded area.
I hope this step-by-step explanation helps you solve the inequality and find the number of people and couples needed for the Spring Dance!