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