Yani's Weekly Hours Over The Course Of The Year Can Be Represented By This Pair Of Inequalities, Where \[$ H \$\] Is The Number Of Hours He Works Each Week:$\[ 8.75h - 10 \ \textless \ 95 \quad \text{or} \quad 8.75h - 10 \geq 305

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Yani's Weekly Hours: A Mathematical Analysis

In this article, we will delve into the world of mathematics and explore a real-world scenario involving inequalities. We will examine the weekly hours worked by Yani, represented by a pair of inequalities, and analyze the possible values of his weekly hours. This discussion will not only provide insight into the mathematical concepts involved but also demonstrate the practical application of inequalities in real-world problems.

The pair of inequalities representing Yani's weekly hours is given by:

8.75h−10<95or8.75h−10≥305{ 8.75h - 10 < 95 \quad \text{or} \quad 8.75h - 10 \geq 305 }

To begin, let's focus on the first inequality:

8.75h−10<95{ 8.75h - 10 < 95 }

We can start by isolating the variable h{h} by adding 10 to both sides of the inequality:

8.75h<105{ 8.75h < 105 }

Next, we can divide both sides of the inequality by 8.75 to solve for h{h}:

h<1058.75{ h < \frac{105}{8.75} }

Using a calculator, we can evaluate the expression on the right-hand side:

h<12{ h < 12 }

This tells us that the value of h{h} must be less than 12.

Now, let's turn our attention to the second inequality:

8.75h−10≥305{ 8.75h - 10 \geq 305 }

We can start by adding 10 to both sides of the inequality:

8.75h≥315{ 8.75h \geq 315 }

Next, we can divide both sides of the inequality by 8.75 to solve for h{h}:

h≥3158.75{ h \geq \frac{315}{8.75} }

Using a calculator, we can evaluate the expression on the right-hand side:

h≥35.86{ h \geq 35.86 }

This tells us that the value of h{h} must be greater than or equal to 35.86.

Since the two inequalities are connected by the word "or," we can combine them to form a single compound inequality:

8.75h−10<95or8.75h−10≥305{ 8.75h - 10 < 95 \quad \text{or} \quad 8.75h - 10 \geq 305 }

This can be rewritten as:

h<12orh≥35.86{ h < 12 \quad \text{or} \quad h \geq 35.86 }

To visualize the solution to the compound inequality, we can create a number line with the values of h{h} marked on it. We can then shade the regions that satisfy the inequality.

The number line will have two shaded regions: one to the left of 12 and one to the right of 35.86.

In this article, we analyzed a pair of inequalities representing Yani's weekly hours and derived the possible values of his weekly hours. We found that the value of h{h} must be less than 12 or greater than or equal to 35.86. This discussion demonstrated the practical application of inequalities in real-world problems and provided insight into the mathematical concepts involved.

The concept of inequalities has numerous real-world applications, including:

  • Finance: Inequalities are used to model financial transactions, such as investments and loans.
  • Science: Inequalities are used to describe physical phenomena, such as the motion of objects and the behavior of particles.
  • Engineering: Inequalities are used to design and optimize systems, such as bridges and buildings.

In the future, we can explore more complex inequalities and their applications in real-world problems. We can also investigate the use of inequalities in machine learning and data analysis.

  • [1] "Inequalities" by Khan Academy
  • [2] "Inequalities" by Math Open Reference
  • [3] "Inequalities" by Wolfram MathWorld
  • Inequality: A statement that two expressions are not equal.
  • Compound inequality: A statement that combines two or more inequalities with the word "or" or "and."
  • Number line: A graphical representation of the real numbers, with points marked at regular intervals.
  • [1] Khan Academy: Inequalities
  • [2] Math Open Reference: Inequalities
  • [3] Wolfram MathWorld: Inequalities
    Yani's Weekly Hours: A Mathematical Analysis - Q&A

In our previous article, we analyzed a pair of inequalities representing Yani's weekly hours and derived the possible values of his weekly hours. We found that the value of h{h} must be less than 12 or greater than or equal to 35.86. In this article, we will answer some frequently asked questions related to the topic.

Q: What is the meaning of the inequalities in the context of Yani's weekly hours?

A: The inequalities represent the possible values of Yani's weekly hours. The first inequality, h<12{h < 12}, indicates that Yani works less than 12 hours per week. The second inequality, h≥35.86{h \geq 35.86}, indicates that Yani works 35.86 hours or more per week.

Q: Why are there two inequalities?

A: The two inequalities are connected by the word "or," which means that Yani's weekly hours can satisfy either of the two conditions. In other words, Yani can work less than 12 hours per week or 35.86 hours or more per week.

Q: How do I solve the inequalities?

A: To solve the inequalities, you can follow these steps:

  1. Isolate the variable h{h} by adding or subtracting the same value from both sides of the inequality.
  2. Divide both sides of the inequality by the coefficient of h{h}.
  3. Evaluate the expression on the right-hand side using a calculator.

Q: What is the significance of the number 8.75 in the inequalities?

A: The number 8.75 is the coefficient of h{h} in the inequalities. It represents the number of hours Yani works per week.

Q: Can I use the inequalities to determine Yani's weekly hours?

A: Yes, you can use the inequalities to determine Yani's weekly hours. However, you need to consider the context of the problem and the possible values of Yani's weekly hours.

Q: How do I graph the inequalities on a number line?

A: To graph the inequalities on a number line, you can follow these steps:

  1. Draw a number line with the values of h{h} marked on it.
  2. Shade the regions that satisfy the inequality.
  3. Draw a closed circle at the endpoint of the inequality.

Q: What are some real-world applications of inequalities?

A: Inequalities have numerous real-world applications, including finance, science, and engineering. They are used to model financial transactions, describe physical phenomena, and design and optimize systems.

Q: Can I use inequalities to solve other types of problems?

A: Yes, you can use inequalities to solve other types of problems, such as optimization problems and decision-making problems.

In this article, we answered some frequently asked questions related to the topic of Yani's weekly hours and inequalities. We hope that this Q&A article has provided you with a better understanding of the concept of inequalities and their applications in real-world problems.

  • [1] Khan Academy: Inequalities
  • [2] Math Open Reference: Inequalities
  • [3] Wolfram MathWorld: Inequalities
  • Inequality: A statement that two expressions are not equal.
  • Compound inequality: A statement that combines two or more inequalities with the word "or" or "and."
  • Number line: A graphical representation of the real numbers, with points marked at regular intervals.
  • Solve the inequality 2x+5<11{2x + 5 < 11}.
  • Graph the inequality x>3{x > 3} on a number line.
  • Use inequalities to determine the possible values of a variable in a real-world problem.
  • x<3{x < 3}
  • The region to the right of 3 is shaded.
  • The possible values of the variable are 3 or more.