The Inequality $|z-4|\ \textless \ |z-2|$ Represents The Region Given By:

The Inequality $|z-4|\ \textless \ |z-2|$ Represents the Region Given by

In mathematics, inequalities involving absolute values are used to describe regions in the complex plane. The inequality $|z-4|\ \textless \ |z-2|$ is a specific example of such an inequality, where $z$ is a complex number. In this article, we will explore the region represented by this inequality and provide a detailed explanation of its geometric interpretation.

To begin with, let's understand the inequality $|z-4|\ \textless \ |z-2|$. Here, $z$ is a complex number, which can be represented as $z = x + yi$, where $x$ and $y$ are real numbers, and $i$ is the imaginary unit. The absolute value of a complex number $z$ is defined as $|z| = \sqrt{x^2 + y^2}$.

The inequality $|z-4|\ \textless \ |z-2|$ can be rewritten as $|z-4|^2 \ \textless \ |z-2|^2$. Expanding the squares, we get $(x-4)^2 + y^2 \ \textless \ (x-2)^2 + y^2$.

To simplify the inequality, we can expand the squares and combine like terms. This gives us:

$(x-4)^2 + y^2 \ \textless \ (x-2)^2 + y^2$

Expanding the squares, we get:

$x^2 - 8x + 16 + y^2 \ \textless \ x^2 - 4x + 4 + y^2$

Combining like terms, we get:

$-8x + 16 \ \textless \ -4x + 4$

Subtracting $-4x$ from both sides, we get:

$-4x + 16 \ \textless \ 4$

Subtracting 16 from both sides, we get:

$-4x \ \textless \ -12$

Dividing both sides by -4, we get:

$x \ \textgreater \ 3$

The inequality $x \ \textgreater \ 3$ represents the region to the right of the line $x = 3$. This means that any complex number $z$ that satisfies the inequality $|z-4|\ \textless \ |z-2|$ must lie to the right of the line $x = 3$.

To visualize this region, we can plot the lines $x = 3$ and $x = 4$ on the complex plane. The line $x = 3$ is a vertical line that passes through the point $(3, 0)$, while the line $x = 4$ is a vertical line that passes through the point $(4, 0)$.

The region represented by the inequality $|z-4|\ \textless \ |z-2|$ is the area to the right of the line $x = 3$ and to the left of the line $x = 4$. This region is bounded by the two vertical lines $x = 3$ and $x = 4$, and it extends infinitely in the positive $y$-direction.

In conclusion, the inequality $|z-4|\ \textless \ |z-2|$ represents the region given by the area to the right of the line $x = 3$ and to the left of the line $x = 4$. This region is bounded by the two vertical lines $x = 3$ and $x = 4$, and it extends infinitely in the positive $y$-direction.

To visualize the region represented by the inequality, we can plot the lines $x = 3$ and $x = 4$ on the complex plane. The line $x = 3$ is a vertical line that passes through the point $(3, 0)$, while the line $x = 4$ is a vertical line that passes through the point $(4, 0)$.

The region represented by the inequality is the area to the right of the line $x = 3$ and to the left of the line $x = 4$. This region is bounded by the two vertical lines $x = 3$ and $x = 4$, and it extends infinitely in the positive $y$-direction.

• The inequality $|z-4|\ \textless \ |z-2|$ represents the region given by the area to the right of the line $x = 3$ and to the left of the line $x = 4$.
• The region represented by the inequality is bounded by the two vertical lines $x = 3$ and $x = 4$, and it extends infinitely in the positive $y$-direction.
• The inequality can be rewritten as $|z-4|^2 \ \textless \ |z-2|^2$, which can be expanded and simplified to $x \ \textgreater \ 3$.

In conclusion, the inequality $|z-4|\ \textless \ |z-2|$ represents the region given by the area to the right of the line $x = 3$ and to the left of the line $x = 4$. This region is bounded by the two vertical lines $x = 3$ and $x = 4$, and it extends infinitely in the positive $y$-direction. The inequality can be rewritten as $|z-4|^2 \ \textless \ |z-2|^2$, which can be expanded and simplified to $x \ \textgreater \ 3$.
Q&A: The Inequality $|z-4|\ \textless \ |z-2|$ Represents the Region Given by

We have received many questions about the inequality $|z-4|\ \textless \ |z-2|$ and the region it represents. Here are some of the most frequently asked questions and their answers:

Q: What is the region represented by the inequality $|z-4|\ \textless \ |z-2|$? A: The region represented by the inequality $|z-4|\ \textless \ |z-2|$ is the area to the right of the line $x = 3$ and to the left of the line $x = 4$. This region is bounded by the two vertical lines $x = 3$ and $x = 4$, and it extends infinitely in the positive $y$-direction.

Q: How do I visualize the region represented by the inequality? A: To visualize the region represented by the inequality, you can plot the lines $x = 3$ and $x = 4$ on the complex plane. The line $x = 3$ is a vertical line that passes through the point $(3, 0)$, while the line $x = 4$ is a vertical line that passes through the point $(4, 0)$. The region represented by the inequality is the area to the right of the line $x = 3$ and to the left of the line $x = 4$.

Q: What is the significance of the inequality $|z-4|\ \textless \ |z-2|$? A: The inequality $|z-4|\ \textless \ |z-2|$ is significant because it represents a region in the complex plane that is bounded by two vertical lines. This region is of interest in mathematics and has many applications in fields such as physics and engineering.

Q: Can I use the inequality $|z-4|\ \textless \ |z-2|$ to solve problems in mathematics? A: Yes, you can use the inequality $|z-4|\ \textless \ |z-2|$ to solve problems in mathematics. For example, you can use it to find the area of the region represented by the inequality, or to determine the distance between two points in the complex plane.

Q: How do I simplify the inequality $|z-4|\ \textless \ |z-2|$? A: To simplify the inequality $|z-4|\ \textless \ |z-2|$, you can expand the squares and combine like terms. This gives you the inequality $x \ \textgreater \ 3$, which is a much simpler expression.

Q: What is the relationship between the inequality $|z-4|\ \textless \ |z-2|$ and the line $x = 3$? A: The inequality $|z-4|\ \textless \ |z-2|$ is related to the line $x = 3$ because the region represented by the inequality is bounded by the two vertical lines $x = 3$ and $x = 4$. The line $x = 3$ is a boundary of the region, and any point to the right of this line satisfies the inequality.

Q: Can I use the inequality $|z-4|\ \textless \ |z-2|$ to solve problems in physics and engineering? A: Yes, you can use the inequality $|z-4|\ \textless \ |z-2|$ to solve problems in physics and engineering. For example, you can use it to model the behavior of a physical system that is bounded by two vertical lines.

In conclusion, the inequality $|z-4|\ \textless \ |z-2|$ represents a region in the complex plane that is bounded by two vertical lines. This region is of interest in mathematics and has many applications in fields such as physics and engineering. We hope that this Q&A article has been helpful in answering your questions about the inequality and the region it represents.

If you are interested in learning more about the inequality $|z-4|\ \textless \ |z-2|$ and the region it represents, we recommend the following resources:

• A textbook on complex analysis that covers the topic of inequalities in the complex plane.
• A research paper on the application of the inequality $|z-4|\ \textless \ |z-2|$ in physics and engineering.
• A online course on mathematics that covers the topic of inequalities in the complex plane.

We hope that these resources are helpful in your studies and research.