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To solve the inequality \( \sqrt{4x + 1} \leq 5.83 \), we first square both sides to eliminate the square root, which gives us:
\[
4x + 1 \leq (5.83)^2
\]
Calculating \( (5.83)^2 \):
\[
(5.83)^2 = 33.9889
\]
Now we rewrite the inequality:
\[
4x + 1 \leq 33.9889
\]
Next, we subtract 1 from both sides:
\[
4x \leq 32.9889
\]
Now, we divide by 4:
\[
x \leq \frac{32.9889}{4}
\]
Calculating this gives:
\[
x \leq 8.247225
\]
So, the values for \( x \) that satisfy the inequality \( \sqrt{4x + 1} \leq 5.83 \) must be \( x \leq 8.247225 \).
Among the provided options (10, 9, 9.5, 8), the only value that satisfies this inequality is:
\[
\boxed{8}
\]