Find a point on the boundary line

Oct 13, 2025 by windailys.com 92 views

Introduction

In mathematics, inequalities are a fundamental concept that helps us understand the relationships between different variables. One of the most common types of inequalities is the linear inequality, which is represented in the form of ax + by ≥ c, where a, b, and c are constants. In this article, we will focus on solving linear inequalities and understanding test points.

What are Test Points?

A test point is a point on the boundary line of the inequality that helps us determine whether the inequality is true or false. In other words, it is a point that we can use to test the inequality. To find a test point, we need to find a point on the boundary line that satisfies the inequality.

How to Find a Test Point

To find a test point, we need to follow these steps:

Write the inequality in the form of ax + by ≥ c: The first step is to write the inequality in the form of ax + by ≥ c. This will help us identify the boundary line.
Find the boundary line: The boundary line is the line that separates the region where the inequality is true from the region where the inequality is false. To find the boundary line, we need to set the inequality to an equality, i.e., ax + by = c.
Find a point on the boundary line: Once we have the boundary line, we need to find a point on it. This point will be our test point.
Test the inequality: Finally, we need to test the inequality using the test point. If the inequality is true, then the test point is a valid solution.

Solving the Inequality

Now, let's solve the inequality $\frac{2}{5} x - \frac{2}{5} y \geq 0$ . To solve this inequality, we need to follow the steps outlined above.

Step 1: Write the inequality in the form of ax + by ≥ c

The given inequality is already in the form of ax + by ≥ c, where a = $\frac{2}{5}$ , b = - $\frac{2}{5}$ , and c = 0.

Step 2: Find the boundary line

To find the boundary line, we need to set the inequality to an equality, i.e., $\frac{2}{5} x - \frac{2}{5} y = 0$ . This can be rewritten as $y = \frac{1}{2} x$ .

Step 3: Find a point on the boundary line

To find a point on the boundary line, we can substitute a value of x into the equation $y = \frac{1}{2} x$ . Let's substitute x = 2 into the equation. This gives us y = 1.

Step 4: Test the inequality

Now, let's test the inequality using the test point (2, 1). We can substitute x = 2 and y = 1 into the original inequality: $\frac{2}{5} (2) - \frac{2}{5} (1) \geq 0$ . This simplifies to $\frac{4}{5} - \frac{2}{5} \geq 0$ , which is true.

Conclusion

In conclusion, the test point (2, 1) holds true for the inequality $\frac{2}{5} x - \frac{2}{5} y \geq 0$ . This means that the point (2, 1) is a valid solution to the inequality.

Comparing Test Points

Now, let's compare the test point (2, 1) with the other options:

Option A: (5, 15): This point does not satisfy the inequality, as $\frac{2}{5} (5) - \frac{2}{5} (15) < 0$ .
Option B: (1/2, 5): This point does not satisfy the inequality, as $\frac{2}{5} (1/2) - \frac{2}{5} (5) < 0$ .
Option C: (5/2, 5): This point does not satisfy the inequality, as $\frac{2}{5} (5/2) - \frac{2}{5} (5) < 0$ .
Option D: (2, 5): This point does not satisfy the inequality, as $\frac{2}{5} (2) - \frac{2}{5} (5) < 0$ .
Option E: (-1, 0): This point does not satisfy the inequality, as $\frac{2}{5} (-1) - \frac{2}{5} (0) < 0$ .

Conclusion

In conclusion, the test point (2, 1) is the only option that holds true for the inequality $\frac{2}{5} x - \frac{2}{5} y \geq 0$ . This means that the point (2, 1) is a valid solution to the inequality.

Final Answer

The final answer is (2, 1).

Introduction

In our previous article, we discussed how to solve linear inequalities and understand test points. In this article, we will answer some frequently asked questions related to solving linear inequalities and understanding test points.

Q: What is a test point?

A: A test point is a point on the boundary line of the inequality that helps us determine whether the inequality is true or false. In other words, it is a point that we can use to test the inequality.

Q: How do I find a test point?

A: To find a test point, you need to follow these steps:

Write the inequality in the form of ax + by ≥ c.
Find the boundary line by setting the inequality to an equality, i.e., ax + by = c.
Find a point on the boundary line.
Test the inequality using the test point.

Q: What is the boundary line?

A: The boundary line is the line that separates the region where the inequality is true from the region where the inequality is false. To find the boundary line, you need to set the inequality to an equality, i.e., ax + by = c.

Q: How do I test the inequality?

A: To test the inequality, you need to substitute the test point into the original inequality. If the inequality is true, then the test point is a valid solution.

Q: What if the test point does not satisfy the inequality?

A: If the test point does not satisfy the inequality, then it is not a valid solution. You need to find another test point and repeat the process.

Q: Can I use any point on the boundary line as a test point?

A: No, you cannot use any point on the boundary line as a test point. You need to find a point that satisfies the inequality.

Q: How do I know if the test point is a valid solution?

A: To know if the test point is a valid solution, you need to test the inequality using the test point. If the inequality is true, then the test point is a valid solution.

Q: Can I use a test point to solve a system of linear inequalities?

A: Yes, you can use a test point to solve a system of linear inequalities. However, you need to find a test point that satisfies all the inequalities in the system.

Q: What if I have a system of linear inequalities with multiple variables?

A: If you have a system of linear inequalities with multiple variables, you need to find a test point that satisfies all the inequalities in the system. You can use the same steps as before to find a test point.

Q: Can I use a test point to solve a linear inequality with fractions?

A: Yes, you can use a test point to solve a linear inequality with fractions. However, you need to follow the same steps as before to find a test point.

Conclusion

In conclusion, solving linear inequalities and understanding test points is an important concept in mathematics. By following the steps outlined in this article, you can solve linear inequalities and understand test points.

Final Answer

The final answer is that solving linear inequalities and understanding test points is an important concept in mathematics that can be used to solve a wide range of problems.

Additional Resources

If you want to learn more about solving linear inequalities and understanding test points, you can check out the following resources:

Khan Academy: Solving Linear Inequalities
Mathway: Solving Linear Inequalities
Wolfram Alpha: Solving Linear Inequalities

Conclusion

In conclusion, solving linear inequalities and understanding test points is an important concept in mathematics that can be used to solve a wide range of problems. By following the steps outlined in this article, you can solve linear inequalities and understand test points.