Question
For what values of \( x \) is the following expression a real number? \( \sqrt{x-7} \)
Ask by Schultz Welch. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( x \) must be greater than or equal to 7 for \( \sqrt{x-7} \) to be a real number.
Solution
To determine the values of \( x \) for which the expression \( \sqrt{x-7} \) is a real number, we need to consider the properties of square roots. The expression under the square root must be non-negative (greater than or equal to zero) for the square root to yield a real number.
Thus, we set up the inequality:
\[
x - 7 \geq 0
\]
Now, we can solve this inequality step by step:
1. Add 7 to both sides of the inequality:
\[
x \geq 7
\]
This means that the expression \( \sqrt{x-7} \) is a real number for all values of \( x \) that are greater than or equal to 7.
In conclusion, the values of \( x \) for which \( \sqrt{x-7} \) is a real number are:
\[
x \geq 7
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution The Deep Dive
For the expression \( \sqrt{x-7} \) to be a real number, the value inside the square root must be non-negative. This means the inequality \( x - 7 \geq 0 \) must hold true. Solving this gives \( x \geq 7 \). Therefore, any value of \( x \) that is 7 or greater will result in a real number for the expression. Additionally, if you think about it, the square root function only produces real results when we have non-negative arguments, so grappling with this concept helps to firmly understand real-valued functions. Next time you're working with square roots, remember to check that the expression under the radical is positive or zero—it's a fun little safety check!
