Ezra Works Two Summer Jobs To Save For A Laptop That Costs At Least $\$1100$. He Charges $\$15/\text{hr}$ To Mow Lawns And $\$10/\text{hr}$ To Walk Dogs. The Inequality That Represents This Situation Is:$\[ 15x + 10y \geq

by ADMIN 222 views

Ezra's Summer Jobs: A Mathematical Inequality

Ezra is a hardworking individual who has taken on two summer jobs to save up for a laptop that costs at least $1100\$1100. He earns $15/hr\$15/\text{hr} by mowing lawns and $10/hr\$10/\text{hr} by walking dogs. In this article, we will explore the mathematical inequality that represents Ezra's situation and how it can be used to determine the minimum number of hours he needs to work to reach his goal.

The inequality that represents Ezra's situation is given by:

15x+10y≥1100{ 15x + 10y \geq 1100 }

where xx represents the number of hours Ezra spends mowing lawns and yy represents the number of hours he spends walking dogs.

To understand the inequality, let's break it down into its components. The left-hand side of the inequality represents the total amount of money Ezra earns from his two jobs, which is given by the sum of the money he earns from mowing lawns (15x15x) and the money he earns from walking dogs (10y10y). The right-hand side of the inequality represents the minimum amount of money Ezra needs to earn to buy the laptop, which is $1100\$1100.

To solve the inequality, we need to find the minimum values of xx and yy that satisfy the inequality. We can do this by isolating the variables on one side of the inequality.

First, let's isolate the variable xx by subtracting 10y10y from both sides of the inequality:

15x≥1100−10y{ 15x \geq 1100 - 10y }

Next, let's divide both sides of the inequality by 1515 to solve for xx:

x≥1100−10y15{ x \geq \frac{1100 - 10y}{15} }

Now that we have the inequality in terms of xx, we can use it to find the minimum number of hours Ezra needs to work to reach his goal. Let's assume that Ezra wants to work a certain number of hours, say xx hours mowing lawns and yy hours walking dogs. We can substitute these values into the inequality to get:

15x+10y≥1100{ 15x + 10y \geq 1100 }

Substituting x=40x = 40 and y=0y = 0, we get:

15(40)+10(0)≥1100{ 15(40) + 10(0) \geq 1100 }

Simplifying the inequality, we get:

600≥1100{ 600 \geq 1100 }

This is not true, so we need to increase the number of hours Ezra works. Let's try x=50x = 50 and y=0y = 0:

15(50)+10(0)≥1100{ 15(50) + 10(0) \geq 1100 }

Simplifying the inequality, we get:

750≥1100{ 750 \geq 1100 }

This is still not true, so we need to increase the number of hours Ezra works again. Let's try x=60x = 60 and y=0y = 0:

15(60)+10(0)≥1100{ 15(60) + 10(0) \geq 1100 }

Simplifying the inequality, we get:

900≥1100{ 900 \geq 1100 }

This is still not true, so we need to increase the number of hours Ezra works again. Let's try x=70x = 70 and y=0y = 0:

15(70)+10(0)≥1100{ 15(70) + 10(0) \geq 1100 }

Simplifying the inequality, we get:

1050≥1100{ 1050 \geq 1100 }

This is still not true, so we need to increase the number of hours Ezra works again. Let's try x=80x = 80 and y=0y = 0:

15(80)+10(0)≥1100{ 15(80) + 10(0) \geq 1100 }

Simplifying the inequality, we get:

1200≥1100{ 1200 \geq 1100 }

This is true, so we have found the minimum number of hours Ezra needs to work to reach his goal.

In conclusion, the inequality 15x+10y≥110015x + 10y \geq 1100 represents Ezra's situation and can be used to determine the minimum number of hours he needs to work to reach his goal. By solving the inequality, we found that Ezra needs to work at least 8080 hours mowing lawns to earn the minimum amount of money needed to buy the laptop.

The inequality 15x+10y≥110015x + 10y \geq 1100 has many real-world applications. For example, it can be used to determine the minimum number of hours a person needs to work to earn a certain amount of money, or to find the minimum number of hours a business needs to operate to meet its financial goals.

In the future, we can use the inequality 15x+10y≥110015x + 10y \geq 1100 to explore other real-world applications, such as determining the minimum number of hours a person needs to work to earn a certain amount of money while also considering other factors such as taxes and expenses.

  • [1] "Algebra and Trigonometry" by Michael Sullivan
  • [2] "Mathematics for the Nonmathematician" by Morris Kline

The following is a list of the variables and constants used in the inequality:

  • xx: number of hours Ezra spends mowing lawns
  • yy: number of hours Ezra spends walking dogs
  • 1515: hourly wage for mowing lawns
  • 1010: hourly wage for walking dogs
  • 11001100: minimum amount of money Ezra needs to earn to buy the laptop
    Ezra's Summer Jobs: A Mathematical Inequality - Q&A

In our previous article, we explored the mathematical inequality that represents Ezra's situation of working two summer jobs to save up for a laptop that costs at least $1100\$1100. We solved the inequality and found that Ezra needs to work at least 8080 hours mowing lawns to earn the minimum amount of money needed to buy the laptop. In this article, we will answer some frequently asked questions about Ezra's situation and the inequality.

A: The hourly wage for mowing lawns is $15\$15.

A: The hourly wage for walking dogs is $10\$10.

A: Ezra needs to work at least 8080 hours mowing lawns to earn the minimum amount of money needed to buy the laptop.

A: The minimum amount of money Ezra needs to earn to buy the laptop is $1100\$1100.

A: No, Ezra cannot earn the minimum amount of money needed to buy the laptop by only working one job. He needs to work both jobs to earn the required amount.

A: Ezra can use the inequality 15x+10y≥110015x + 10y \geq 1100 to determine the minimum number of hours he needs to work to earn the minimum amount of money needed to buy the laptop. He can substitute different values of xx and yy into the inequality to find the minimum number of hours he needs to work.

A: The inequality 15x+10y≥110015x + 10y \geq 1100 has many real-world applications, such as determining the minimum number of hours a person needs to work to earn a certain amount of money, or to find the minimum number of hours a business needs to operate to meet its financial goals.

A: Yes, the inequality can be used to determine the minimum number of hours a person needs to work to earn a certain amount of money while also considering other factors such as taxes and expenses. However, this would require modifying the inequality to take into account these additional factors.

In conclusion, the inequality 15x+10y≥110015x + 10y \geq 1100 represents Ezra's situation and can be used to determine the minimum number of hours he needs to work to earn the minimum amount of money needed to buy the laptop. We have answered some frequently asked questions about Ezra's situation and the inequality, and have discussed some real-world applications of the inequality.

The inequality 15x+10y≥110015x + 10y \geq 1100 has many real-world applications, such as:

  • Determining the minimum number of hours a person needs to work to earn a certain amount of money
  • Finding the minimum number of hours a business needs to operate to meet its financial goals
  • Determining the minimum number of hours a person needs to work to earn a certain amount of money while also considering other factors such as taxes and expenses

In the future, we can use the inequality 15x+10y≥110015x + 10y \geq 1100 to explore other real-world applications, such as:

  • Determining the minimum number of hours a person needs to work to earn a certain amount of money while also considering other factors such as taxes and expenses
  • Finding the minimum number of hours a business needs to operate to meet its financial goals while also considering other factors such as taxes and expenses
  • [1] "Algebra and Trigonometry" by Michael Sullivan
  • [2] "Mathematics for the Nonmathematician" by Morris Kline

The following is a list of the variables and constants used in the inequality:

  • xx: number of hours Ezra spends mowing lawns
  • yy: number of hours Ezra spends walking dogs
  • 1515: hourly wage for mowing lawns
  • 1010: hourly wage for walking dogs
  • 11001100: minimum amount of money Ezra needs to earn to buy the laptop