In this real-world scenario of raffle ticket sales, let's break down the situation step by step using both equations and inequalities.
1. Understanding the Problem
You want to sell tickets such that:
- You have a target of selling 200 tickets.
- You also have an upper limit of 400 tickets to avoid overselling.
2. Setting Up the Equations
Equation
If you want to express your target for ticket sales as an equation, you would say: \[ T = 200 \] This means the ideal or target number of tickets sold is 200.
However, if you want to express the range of tickets you can sell, an equation may not fully capture the inherent flexibility you desire, since equations imply strict equality.
Inequalities
To represent the possible number of tickets sold, use inequalities: \[ 200 \leq T \leq 400 \] This shows that you want to sell at least 200 tickets but no more than 400 tickets.
3. Include Already Sold Tickets (S)
If you’ve already sold some tickets, represented by the variable \( S \), you want to ensure the total number of tickets sold \( T \) does not exceed 400. The total tickets sold can now be expressed using:
\[ S + T \leq 400 \] This inequality states that the sum of the tickets already sold and the target number of tickets sold should not exceed 400.
4. Rearranging the Inequality
If you want \( T \) by itself, rearranging gives: \[ T \leq 400 - S \]
Summary
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To express where you want to be with ticket sales (target): \[ T = 200 \]
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To define your limits on ticket sales (inequality): \[ 200 \leq T \leq 400 \]
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Incorporating already sold tickets: \[ S + T \leq 400 \] or rearranged, \[ T \leq 400 - S \]
Conclusion
In this scenario, the equations and inequalities allow you to clearly define your ticket sales goals and constraints while considering the tickets already sold. This flexible approach helps in planning and ensures you remain within the limits of your raffle ticket sales.
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