Answer :
To solve the inequality [tex]\(\sqrt{x} \geq 9\)[/tex], we need to find the values of [tex]\(x\)[/tex] that satisfy this condition. Here's a step-by-step breakdown of how to determine the correct values:
1. Understand the inequality: The inequality [tex]\(\sqrt{x} \geq 9\)[/tex] means that the square root of [tex]\(x\)[/tex] should be greater than or equal to 9.
2. Square both sides to eliminate the square root:
[tex]\[
(\sqrt{x})^2 \geq 9^2
\][/tex]
[tex]\[
x \geq 81
\][/tex]
Squaring both sides gives us [tex]\(x \geq 81\)[/tex]. This tells us that [tex]\(x\)[/tex] must be at least 81 or greater for the inequality to hold true.
3. Evaluate each given value:
- 27: Since 27 is less than 81, it does not satisfy the inequality.
- 100: Since 100 is greater than 81, it does satisfy the inequality.
- -3: Square root is not defined for negative numbers in the context of real numbers, so [tex]\(-3\)[/tex] does not satisfy the inequality.
- 81: Since 81 is equal to 81, it does satisfy the inequality.
- -81: Similar to [tex]\(-3\)[/tex], the square root of a negative number is not considered here, so [tex]\(-81\)[/tex] does not satisfy the inequality.
- 3: Since 3 is less than 81, it does not satisfy the inequality.
4. Conclusion:
The values that satisfy the inequality [tex]\(\sqrt{x} \geq 9\)[/tex] are 100 and 81. Thus, the solutions are:
[tex]\[ \boxed{100, 81} \][/tex]
1. Understand the inequality: The inequality [tex]\(\sqrt{x} \geq 9\)[/tex] means that the square root of [tex]\(x\)[/tex] should be greater than or equal to 9.
2. Square both sides to eliminate the square root:
[tex]\[
(\sqrt{x})^2 \geq 9^2
\][/tex]
[tex]\[
x \geq 81
\][/tex]
Squaring both sides gives us [tex]\(x \geq 81\)[/tex]. This tells us that [tex]\(x\)[/tex] must be at least 81 or greater for the inequality to hold true.
3. Evaluate each given value:
- 27: Since 27 is less than 81, it does not satisfy the inequality.
- 100: Since 100 is greater than 81, it does satisfy the inequality.
- -3: Square root is not defined for negative numbers in the context of real numbers, so [tex]\(-3\)[/tex] does not satisfy the inequality.
- 81: Since 81 is equal to 81, it does satisfy the inequality.
- -81: Similar to [tex]\(-3\)[/tex], the square root of a negative number is not considered here, so [tex]\(-81\)[/tex] does not satisfy the inequality.
- 3: Since 3 is less than 81, it does not satisfy the inequality.
4. Conclusion:
The values that satisfy the inequality [tex]\(\sqrt{x} \geq 9\)[/tex] are 100 and 81. Thus, the solutions are:
[tex]\[ \boxed{100, 81} \][/tex]
