This System Of Linear Inequalities Can Be Used To Find The Possible Heights, In Inches, Of Darius, \[$d\$\], And His Brother William, \[$w\$\].$\[ \begin{array}{l} d \geq 36 \\ w \ \textless \ 68 \\ d \leq 4 +

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Introduction

In mathematics, systems of linear inequalities are used to represent a set of conditions that a variable or variables must satisfy. These conditions can be used to find the possible values of the variables, which is a crucial aspect of problem-solving in various fields. In this article, we will explore a system of linear inequalities that can be used to find the possible heights of Darius and his brother William.

The Problem

The problem states that we need to find the possible heights of Darius and William in inches, represented by the variables dd and ww respectively. The system of linear inequalities is given as:

d≥36w \textless 68d≤4+\begin{array}{l} d \geq 36 \\ w \ \textless \ 68 \\ d \leq 4 + \end{array}

Understanding the Inequalities

Let's break down each inequality and understand what it represents.

  • The first inequality, d≥36d \geq 36, states that Darius's height must be greater than or equal to 36 inches.
  • The second inequality, w \textless 68w \ \textless \ 68, states that William's height must be less than 68 inches.
  • The third inequality, d≤4+wd \leq 4 + w, states that Darius's height must be less than or equal to 4 inches more than William's height.

Graphing the Inequalities

To visualize the solution, we can graph the inequalities on a coordinate plane. The first inequality, d≥36d \geq 36, can be graphed as a vertical line at x=36x = 36. The second inequality, w \textless 68w \ \textless \ 68, can be graphed as a vertical line at x=68x = 68, but since it's a strict inequality, we will use an open circle to represent it.

The third inequality, d≤4+wd \leq 4 + w, can be graphed as a line with a slope of 1 and a y-intercept of 4. Since it's a less than or equal to inequality, we will use a closed circle to represent it.

Finding the Solution

To find the solution, we need to find the region where all three inequalities are satisfied. This region will be the intersection of the three graphs.

From the graph, we can see that the solution is the region below the line d=4+wd = 4 + w and above the line d=36d = 36. The region is also bounded by the line w=68w = 68.

Conclusion

In conclusion, the system of linear inequalities can be used to find the possible heights of Darius and William. The solution is the region where all three inequalities are satisfied, which is the intersection of the three graphs. This problem demonstrates the importance of systems of linear inequalities in mathematics and how they can be used to solve real-world problems.

Example Use Cases

This system of linear inequalities can be used in various real-world scenarios, such as:

  • Height restrictions: In a building or a structure, there may be height restrictions for certain areas. This system of linear inequalities can be used to find the possible heights of objects or people in those areas.
  • Age restrictions: In a school or a workplace, there may be age restrictions for certain activities or positions. This system of linear inequalities can be used to find the possible ages of individuals in those areas.
  • Weight restrictions: In a vehicle or a machine, there may be weight restrictions for certain loads or cargo. This system of linear inequalities can be used to find the possible weights of objects in those areas.

Tips and Tricks

When solving systems of linear inequalities, it's essential to remember the following tips and tricks:

  • Graph the inequalities: Graphing the inequalities can help visualize the solution and make it easier to find the intersection of the graphs.
  • Use a coordinate plane: Using a coordinate plane can help to visualize the solution and make it easier to find the intersection of the graphs.
  • Check the inequalities: Make sure to check the inequalities carefully to ensure that they are satisfied in the solution region.

Conclusion

Q: What is a system of linear inequalities?

A: A system of linear inequalities is a set of linear inequalities that are combined to form a single system. Each inequality in the system represents a condition that the variables must satisfy.

Q: How do I graph a system of linear inequalities?

A: To graph a system of linear inequalities, you need to graph each inequality separately on a coordinate plane. Then, find the intersection of the graphs to determine the solution region.

Q: What is the difference between a strict inequality and a non-strict inequality?

A: A strict inequality is an inequality that is represented by a strict symbol, such as < or >. A non-strict inequality is an inequality that is represented by a non-strict symbol, such as ≤ or ≥.

Q: How do I find the solution to a system of linear inequalities?

A: To find the solution to a system of linear inequalities, you need to find the region where all the inequalities are satisfied. This region is the intersection of the graphs of the inequalities.

Q: Can I use a system of linear inequalities to solve a problem with multiple variables?

A: Yes, you can use a system of linear inequalities to solve a problem with multiple variables. However, you need to make sure that the inequalities are consistent and that the solution region is valid.

Q: How do I check if a solution is valid?

A: To check if a solution is valid, you need to make sure that it satisfies all the inequalities in the system. You can do this by plugging the solution into each inequality and checking if it is true.

Q: Can I use a system of linear inequalities to solve a problem with negative numbers?

A: Yes, you can use a system of linear inequalities to solve a problem with negative numbers. However, you need to make sure that the inequalities are consistent and that the solution region is valid.

Q: How do I graph a system of linear inequalities with negative numbers?

A: To graph a system of linear inequalities with negative numbers, you need to graph each inequality separately on a coordinate plane. Then, find the intersection of the graphs to determine the solution region.

Q: Can I use a system of linear inequalities to solve a problem with fractions?

A: Yes, you can use a system of linear inequalities to solve a problem with fractions. However, you need to make sure that the inequalities are consistent and that the solution region is valid.

Q: How do I graph a system of linear inequalities with fractions?

A: To graph a system of linear inequalities with fractions, you need to graph each inequality separately on a coordinate plane. Then, find the intersection of the graphs to determine the solution region.

Q: Can I use a system of linear inequalities to solve a problem with decimals?

A: Yes, you can use a system of linear inequalities to solve a problem with decimals. However, you need to make sure that the inequalities are consistent and that the solution region is valid.

Q: How do I graph a system of linear inequalities with decimals?

A: To graph a system of linear inequalities with decimals, you need to graph each inequality separately on a coordinate plane. Then, find the intersection of the graphs to determine the solution region.

Conclusion

In conclusion, a system of linear inequalities is a powerful tool for solving problems with multiple variables and constraints. By understanding how to graph and solve systems of linear inequalities, you can tackle a wide range of problems in mathematics and real-world applications.