Given:${ C=\frac{x}{5}+\sqrt{\frac{-7}{x-1}} }$1. For What Value(s) Of { X $}$ Will { C $}$ Be A Real Number?

Introduction

In mathematics, the concept of real numbers is crucial in solving various equations and inequalities. A real number is a value that can be expressed on the number line, and it is denoted by the symbol ℝ. In this article, we will focus on finding the value(s) of x for which the given equation will result in a real number. The equation is:

${ C=\frac{x}{5}+\sqrt{\frac{-7}{x-1}} }$

Understanding the Equation

To find the value(s) of x for which C will be a real number, we need to analyze the equation and identify the conditions that must be satisfied. The equation involves a square root term, which means that the expression inside the square root must be non-negative. Additionally, the denominator of the fraction cannot be zero.

Condition for the Square Root Term

The square root term is given by:

${ \sqrt{\frac{-7}{x-1}} }$

For this term to be real, the expression inside the square root must be non-negative. This means that:

${ \frac{-7}{x-1} \geq 0 }$

Simplifying this inequality, we get:

${ x-1 \leq 0 }$

Solving for x, we get:

${ x \leq 1 }$

Condition for the Denominator

The denominator of the fraction is given by:

${ x-1 }$

For the fraction to be defined, the denominator cannot be zero. This means that:

${ x-1 \neq 0 }$

Solving for x, we get:

${ x \neq 1 }$

Combining the Conditions

We have two conditions that must be satisfied for C to be a real number:

1. The expression inside the square root must be non-negative: ${ x \leq 1 }$
2. The denominator cannot be zero: ${ x \neq 1 }$

Combining these conditions, we get:

${ x \leq 1 \text{ and } x \neq 1 }$

This can be simplified to:

${ x < 1 }$

Conclusion

In conclusion, for the given equation to result in a real number, the value of x must satisfy the condition ${ x < 1 }$. This means that x can take any value less than 1, but it cannot be equal to 1.

Graphical Representation

To visualize the solution, we can plot the graph of the equation on the number line. The graph will show the values of x for which C is a real number.

Code Implementation

Here is a Python code snippet that implements the solution:

import numpy as np
def find_real_solutions():
# Define the function
def f(x):
return x/5 + np.sqrt(-7/(x-1))
# Define the interval
x = np.linspace(-10, 0, 100)

# Find the values of x for which C is a real number
real_solutions = x[x < 1]

return real_solutions

real_solutions = find_real_solutions()
print(real_solutions)

This code snippet uses the NumPy library to generate an array of values for x and then finds the values of x for which C is a real number. The result is an array of values that satisfy the condition ${ x < 1 }$.

Final Thoughts

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Introduction

In our previous article, we discussed the concept of real numbers and how to find the value(s) of x for which the given equation will result in a real number. We analyzed the equation, identified the conditions that must be satisfied, and combined the conditions to find the final solution. In this article, we will provide a Q&A section to address any questions or concerns that readers may have.

Q: What is the significance of the square root term in the equation?

A: The square root term is crucial in the equation because it determines whether the expression inside the square root is non-negative. If the expression is non-negative, the square root term will be real, and the equation will have a real solution.

Q: Why is the denominator of the fraction important?

A: The denominator of the fraction is important because it cannot be zero. If the denominator is zero, the fraction will be undefined, and the equation will not have a real solution.

Q: How do we combine the conditions to find the final solution?

A: To combine the conditions, we need to find the intersection of the two conditions. In this case, the conditions are:

1. The expression inside the square root must be non-negative: ${ x \leq 1 }$
2. The denominator cannot be zero: ${ x \neq 1 }$

Combining these conditions, we get:

${ x \leq 1 \text{ and } x \neq 1 }$

This can be simplified to:

${ x < 1 }$

Q: What is the graphical representation of the solution?

A: The graphical representation of the solution is a number line that shows the values of x for which C is a real number. The number line will have a vertical line at x = 1, and all values of x to the left of this line will be included in the solution.

Q: How do we implement the solution in code?

A: We can implement the solution in code using a programming language such as Python. The code will generate an array of values for x and then find the values of x for which C is a real number. The result will be an array of values that satisfy the condition ${ x < 1 }$.

Q: What are some common mistakes to avoid when solving equations with square roots?

A: Some common mistakes to avoid when solving equations with square roots include:

• Not checking if the expression inside the square root is non-negative
• Not considering the case where the denominator is zero
• Not combining the conditions correctly to find the final solution

Q: How do we extend the solution to more complex equations?

A: To extend the solution to more complex equations, we need to analyze the equation and identify the conditions that must be satisfied. We can then combine the conditions to find the final solution. In some cases, we may need to use advanced mathematical techniques such as calculus or linear algebra to solve the equation.

Q: What are some real-world applications of the solution?

A: The solution has many real-world applications in fields such as physics, engineering, and economics. For example, the solution can be used to model population growth, electrical circuits, and financial markets.

Conclusion

In conclusion, the solution to the given equation is ${ x < 1 }$, which means that x can take any value less than 1, but it cannot be equal to 1. We have provided a Q&A section to address any questions or concerns that readers may have. We have also discussed the significance of the square root term, the importance of the denominator, and how to combine the conditions to find the final solution.