Which of the following values for x makes the inequality square root 4x + 1 < (line underneath) 5.83 true?

To solve the inequality \(\sqrt{4x + 1} < 5.83\), we first square both sides to eliminate the square root. This gives us:

\[
4x + 1 < (5.83)^2
\]

Calculating \( (5.83)^2 \):

\[
5.83^2 = 33.9889
\]

Now, we updated our inequality:

\[
4x + 1 < 33.9889
\]

Next, we subtract 1 from both sides:

\[
4x < 33.9889 - 1
\]
\[
4x < 32.9889
\]

Now divide both sides by 4:

\[
x < \frac{32.9889}{4}
\]
\[
x < 8.247225
\]

Now we will check which of the proposed values for \(x\) (9, 9.5, 8, and 10) satisfies the inequality \(x < 8.247225\):

- For \(x = 9\): \(9 \not< 8.247225\)
- For \(x = 9.5\): \(9.5 \not< 8.247225\)
- For \(x = 8\): \(8 < 8.247225\) (this is true)
- For \(x = 10\): \(10 \not< 8.247225\)

The only value of \(x\) that satisfies the inequality is \(8\).

So, the answer is:

**8**