To solve the inequality [tex]\(\sqrt{x} \leq 11\)[/tex], we follow these steps:
1. Understand the inequality:
[tex]\[\sqrt{x} \leq 11\][/tex]
2. Square both sides to eliminate the square root:
[tex]\[(\sqrt{x})^2 \leq 11^2\][/tex]
Simplifies to:
[tex]\[x \leq 121\][/tex]
Note that squaring both sides is a valid operation because the square function is monotonic (non-decreasing) on the domain of [tex]\(\sqrt{x}\)[/tex], which is [tex]\(x \geq 0\)[/tex].
3. Determine the domain:
Since the square root function is defined when the input is non-negative, we have an additional constraint:
[tex]\[x \geq 0\][/tex]
4. Combine both constraints:
The combined solution set is:
[tex]\[0 \leq x \leq 121\][/tex]
5. Express the solution in the required format:
[tex]\[0 =< x \leq 121\][/tex]
So, the final answer is:
[tex]\[0 =< x =< 121\][/tex]
