Conditions for the Square Roots

Oct 13, 2025 by miodeya.com 100 views

**For What Values of x is the Expression Defined?**

Understanding the Problem

The given expression involves square roots, which means we need to consider the values of x that make the expression defined. In other words, we need to find the values of x that satisfy the conditions for the square roots to be real numbers.

The Expression

The given expression is:

$\frac{\sqrt{x+5}}{\sqrt{1-x}}$

Conditions for the Square Roots

For the square roots to be real numbers, the expressions inside the square roots must be non-negative. This means:

$x + 5 \geq 0$
$1 - x \geq 0$

Solving the Inequalities

Let's solve the inequalities:

$x + 5 \geq 0$ $x \geq -5$
$1 - x \geq 0$ $1 \geq x$

Combining the Inequalities

To find the values of x that satisfy both inequalities, we need to combine them:

$-5 \leq x \leq 1$

Conclusion

The expression is defined for values of x that satisfy the inequality:

$-5 \leq x \leq 1$

Q&A

Q: What is the condition for the square roots to be real numbers? A: The expressions inside the square roots must be non-negative.

Q: What are the values of x that satisfy the inequality $x + 5 \geq 0$ ? A: $x \geq -5$

Q: What are the values of x that satisfy the inequality $1 - x \geq 0$ ? A: $1 \geq x$

Q: How do we combine the inequalities to find the values of x that satisfy both? A: We combine the inequalities by finding the intersection of the two sets of values.

Q: What is the final answer to the problem? A: The expression is defined for values of x that satisfy the inequality $-5 \leq x \leq 1$ .

Common Mistakes

Not considering the condition for the square roots to be real numbers.
Not solving the inequalities correctly.
Not combining the inequalities correctly.

Tips and Tricks

Always consider the condition for the square roots to be real numbers.
Solve the inequalities step by step.
Combine the inequalities by finding the intersection of the two sets of values.

Real-World Applications

The concept of square roots and inequalities is used in various real-world applications, such as finance, engineering, and physics.
Understanding the conditions for square roots to be real numbers is crucial in these applications.

Conclusion

In conclusion, the expression is defined for values of x that satisfy the inequality $-5 \leq x \leq 1$ . Understanding the conditions for square roots to be real numbers is crucial in solving this problem. By following the steps outlined in this article, you can solve similar problems and apply the concept to real-world applications.