The Soccer Team Is Conducting A Fundraiser Selling Long-sleeved T-shirts For $\$14$ And Short-sleeved T-shirts For $\$10$. So Far, The Team Has Sold Less Than $\$200$ Worth Of The Two Types Of T-shirts.Which Inequality Best

Introduction

The soccer team is conducting a fundraiser by selling long-sleeved T-shirts for $\$14$ and short-sleeved T-shirts for $\$10$. The team has sold less than $\$200$ worth of the two types of T-shirts. In this scenario, we need to determine which inequality best represents the situation. To do this, we'll consider the number of long-sleeved and short-sleeved T-shirts sold.

Let's Define the Variables

Let's define the variables to represent the number of long-sleeved and short-sleeved T-shirts sold. Let $x$ be the number of long-sleeved T-shirts sold and $y$ be the number of short-sleeved T-shirts sold.

The Cost of T-Shirts Sold

The cost of long-sleeved T-shirts sold is $\$14x$ and the cost of short-sleeved T-shirts sold is $\$10y$. The total cost of T-shirts sold is the sum of these two costs, which is $\$14x + \$10y$.

The Total Cost is Less Than $\$200$

Since the team has sold less than $\$200$ worth of the two types of T-shirts, we can write the inequality:

$$14x + 10y < 200$$

Simplifying the Inequality

We can simplify the inequality by dividing both sides by $2$. This gives us:

$$7x + 5y < 100$$

Conclusion

The inequality $7x + 5y < 100$ best represents the situation. This inequality shows that the total cost of long-sleeved and short-sleeved T-shirts sold is less than $\$100$. We can use this inequality to determine the possible combinations of long-sleeved and short-sleeved T-shirts that can be sold.

Graphing the Inequality

To visualize the inequality, we can graph it on a coordinate plane. The graph of the inequality is a line with a slope of $-\frac{5}{7}$ and a y-intercept of $20$. The inequality is satisfied by all points below this line.

Example

Suppose the soccer team sells $x = 5$ long-sleeved T-shirts and $y = 3$ short-sleeved T-shirts. The total cost of T-shirts sold is:

$$14(5) + 10(3) = 70 + 30 = 100$$

This is not less than $\$200$, so the team cannot sell $x = 5$ long-sleeved T-shirts and $y = 3$ short-sleeved T-shirts.

Another Example

Suppose the soccer team sells $x = 2$ long-sleeved T-shirts and $y = 4$ short-sleeved T-shirts. The total cost of T-shirts sold is:

$$14(2) + 10(4) = 28 + 40 = 68$$

This is less than $\$200$, so the team can sell $x = 2$ long-sleeved T-shirts and $y = 4$ short-sleeved T-shirts.

Conclusion

In conclusion, the inequality $7x + 5y < 100$ best represents the situation. This inequality shows that the total cost of long-sleeved and short-sleeved T-shirts sold is less than $\$100$. We can use this inequality to determine the possible combinations of long-sleeved and short-sleeved T-shirts that can be sold.

Final Answer

The final answer is $\boxed{7x + 5y < 100}$.

Introduction

In our previous article, we discussed the soccer team fundraiser inequality, which is represented by the inequality $7x + 5y < 100$. In this article, we'll answer some frequently asked questions about the inequality and provide additional insights.

Q: What is the significance of the inequality $7x + 5y < 100$?

A: The inequality $7x + 5y < 100$ represents the total cost of long-sleeved and short-sleeved T-shirts sold by the soccer team. The inequality shows that the total cost is less than $\$100$.

Q: What is the slope of the line represented by the inequality $7x + 5y < 100$?

A: The slope of the line represented by the inequality $7x + 5y < 100$ is $-\frac{5}{7}$. This means that for every 7 units increase in the number of long-sleeved T-shirts sold, the number of short-sleeved T-shirts sold decreases by 5 units.

Q: What is the y-intercept of the line represented by the inequality $7x + 5y < 100$?

A: The y-intercept of the line represented by the inequality $7x + 5y < 100$ is 20. This means that if the soccer team sells 0 long-sleeved T-shirts, they can sell up to 20 short-sleeved T-shirts to reach a total cost of $\$100$.

Q: How can I use the inequality $7x + 5y < 100$ to determine the possible combinations of long-sleeved and short-sleeved T-shirts that can be sold?

A: To determine the possible combinations of long-sleeved and short-sleeved T-shirts that can be sold, you can graph the inequality on a coordinate plane. The inequality is satisfied by all points below the line represented by the inequality. You can also use the inequality to find the maximum number of long-sleeved and short-sleeved T-shirts that can be sold.

Q: What is the maximum number of long-sleeved T-shirts that can be sold?

A: To find the maximum number of long-sleeved T-shirts that can be sold, you can set $y = 0$ in the inequality $7x + 5y < 100$. This gives you the inequality $7x < 100$, which simplifies to $x < \frac{100}{7}$. Since $x$ must be an integer, the maximum number of long-sleeved T-shirts that can be sold is 14.

Q: What is the maximum number of short-sleeved T-shirts that can be sold?

A: To find the maximum number of short-sleeved T-shirts that can be sold, you can set $x = 0$ in the inequality $7x + 5y < 100$. This gives you the inequality $5y < 100$, which simplifies to $y < 20$. Since $y$ must be an integer, the maximum number of short-sleeved T-shirts that can be sold is 19.

Q: Can the soccer team sell 10 long-sleeved T-shirts and 15 short-sleeved T-shirts?

A: To determine if the soccer team can sell 10 long-sleeved T-shirts and 15 short-sleeved T-shirts, you can substitute $x = 10$ and $y = 15$ into the inequality $7x + 5y < 100$. This gives you the inequality $7(10) + 5(15) < 100$, which simplifies to $70 + 75 < 100$. Since $145 < 100$, the inequality is not satisfied, and the soccer team cannot sell 10 long-sleeved T-shirts and 15 short-sleeved T-shirts.

Conclusion

In conclusion, the inequality $7x + 5y < 100$ represents the total cost of long-sleeved and short-sleeved T-shirts sold by the soccer team. We've answered some frequently asked questions about the inequality and provided additional insights. We hope this article has been helpful in understanding the soccer team fundraiser inequality.

Final Answer

The final answer is $\boxed{7x + 5y < 100}$.