Tony Has $20. He Wants To Buy At Least 4 Snacks. Hot Dogs ( X ) {(x)} ( X ) Are $3 Each. Peanuts { (y)$}$ Are $2 Each.Complete The Inequalities That Represent This Situation.Total Snacks: { \quad X + Y \geq 4$}$Total Price:

Introduction

Tony has a budget of $20 and wants to buy at least 4 snacks. He is considering two options: hot dogs and peanuts. Hot dogs cost $3 each, while peanuts cost $2 each. In this scenario, we need to determine the inequalities that represent Tony's situation. We will focus on the total number of snacks and the total price.

Total Snacks:

The total number of snacks Tony wants to buy is at least 4. This can be represented by the inequality:

$$x + y \geq 4$$

where $x$ is the number of hot dogs and $y$ is the number of peanuts.

Total Price:

The total price of the snacks Tony wants to buy is at most $20. This can be represented by the inequality:

$$3x + 2y \leq 20$$

where $x$ is the number of hot dogs and $y$ is the number of peanuts.

Solving the Inequalities

To solve the inequalities, we need to find the values of $x$ and $y$ that satisfy both inequalities. We can start by solving the first inequality:

$$x + y \geq 4$$

This inequality tells us that the total number of snacks must be at least 4. We can rewrite this inequality as:

$$y \geq 4 - x$$

Now, we can substitute this expression for $y$ into the second inequality:

$$3x + 2(4 - x) \leq 20$$

Simplifying this inequality, we get:

$$3x + 8 - 2x \leq 20$$

Combine like terms:

$$x + 8 \leq 20$$

Subtract 8 from both sides:

$$x \leq 12$$

Now, we can substitute this expression for $x$ into the first inequality:

$$x + y \geq 4$$

Substitute $x \leq 12$:

$$12 + y \geq 4$$

Subtract 12 from both sides:

$$y \geq -8$$

However, since $y$ represents the number of peanuts, it cannot be negative. Therefore, we can conclude that:

$$y \geq 0$$

Graphing the Inequalities

To visualize the solution to the inequalities, we can graph the two inequalities on a coordinate plane. The first inequality is:

$$x + y \geq 4$$

This inequality represents a line with a slope of -1 and a y-intercept of 4. The second inequality is:

$$3x + 2y \leq 20$$

This inequality represents a line with a slope of -3/2 and a y-intercept of 10. The solution to the inequalities is the region below the line $3x + 2y = 20$ and above the line $x + y = 4$.

Conclusion

In this scenario, Tony wants to buy at least 4 snacks with a budget of $20. The inequalities that represent this situation are:

$$x + y \geq 4$$

$$3x + 2y \leq 20$$

By solving these inequalities, we can determine the values of $x$ and $y$ that satisfy both inequalities. The solution to the inequalities is the region below the line $3x + 2y = 20$ and above the line $x + y = 4$.

Discussion

This scenario can be applied to real-life situations where a person has a budget and wants to buy a certain number of items. The inequalities can be used to determine the maximum number of items that can be bought within the budget.

Example Questions

1. If Tony wants to buy at least 5 snacks, what is the new inequality for the total number of snacks?
2. If the price of hot dogs increases to $4 each, what is the new inequality for the total price?
3. If Tony wants to buy at least 3 hot dogs, what is the new inequality for the total number of snacks?

Answer Key

1. $x + y \geq 5$
2. $4x + 2y \leq 20$
3. $x + y \geq 3$

Additional Resources

For more information on solving inequalities, please refer to the following resources:

• Khan Academy: Solving Inequalities
• Mathway: Solving Inequalities
• Wolfram Alpha: Solving Inequalities
  Tony's Snack Shopping Conundrum: Q&A =====================================

Introduction

In our previous article, we explored the scenario of Tony, who wants to buy at least 4 snacks with a budget of $20. We determined the inequalities that represent this situation and solved them to find the values of $x$ and $y$ that satisfy both inequalities. In this article, we will answer some frequently asked questions related to this scenario.

Q&A

Q: What if Tony wants to buy at least 5 snacks?

A: If Tony wants to buy at least 5 snacks, the new inequality for the total number of snacks would be:

$$x + y \geq 5$$

This means that Tony would need to buy at least 5 snacks in total, which could be a combination of hot dogs and peanuts.

Q: What if the price of hot dogs increases to $4 each?

A: If the price of hot dogs increases to $4 each, the new inequality for the total price would be:

$$4x + 2y \leq 20$$

This means that Tony would need to stay within a budget of $20, but with the increased price of hot dogs, he would need to be more careful with his spending.

Q: What if Tony wants to buy at least 3 hot dogs?

A: If Tony wants to buy at least 3 hot dogs, the new inequality for the total number of snacks would be:

$$x + y \geq 3$$

This means that Tony would need to buy at least 3 hot dogs, and the number of peanuts would be determined by the remaining budget.

Q: Can Tony buy only hot dogs?

A: Yes, Tony can buy only hot dogs. In this case, the inequality for the total number of snacks would be:

$$x \geq 4$$

This means that Tony would need to buy at least 4 hot dogs.

Q: Can Tony buy only peanuts?

A: Yes, Tony can buy only peanuts. In this case, the inequality for the total number of snacks would be:

$$y \geq 4$$

This means that Tony would need to buy at least 4 peanuts.

Q: What is the maximum number of hot dogs Tony can buy?

A: The maximum number of hot dogs Tony can buy is 12. This is because the inequality for the total price is:

$$3x + 2y \leq 20$$

Substituting $y = 0$ (since Tony is buying only hot dogs), we get:

$$3x \leq 20$$

Dividing both sides by 3, we get:

$$x \leq 6.67$$

Since $x$ must be an integer, the maximum number of hot dogs Tony can buy is 6.

Q: What is the maximum number of peanuts Tony can buy?

A: The maximum number of peanuts Tony can buy is 10. This is because the inequality for the total price is:

$$3x + 2y \leq 20$$

Substituting $x = 0$ (since Tony is buying only peanuts), we get:

$$2y \leq 20$$

Dividing both sides by 2, we get:

$$y \leq 10$$

Since $y$ must be an integer, the maximum number of peanuts Tony can buy is 10.

Conclusion

In this article, we answered some frequently asked questions related to Tony's snack shopping conundrum. We explored different scenarios, such as buying at least 5 snacks, increasing the price of hot dogs, and buying at least 3 hot dogs. We also determined the maximum number of hot dogs and peanuts Tony can buy.