We want to solve the inequality:
\[ \sqrt{4x + 1} \leq 5.83 \]
First, square both sides to eliminate the square root (keeping in mind that both sides are non-negative since we're dealing with a square root):
\[ 4x + 1 \leq (5.83)^2 \]
Calculating \( (5.83)^2 \):
\[ (5.83)^2 = 33.9889 \]
So the inequality becomes:
\[ 4x + 1 \leq 33.9889 \]
Next, subtract 1 from both sides:
\[ 4x \leq 33.9889 - 1 \] \[ 4x \leq 32.9889 \]
Now divide both sides by 4:
\[ x \leq \frac{32.9889}{4} \] \[ x \leq 8.247225 \]
Now we can check the provided values: 9.5, 8, 10, and 9 to see which satisfies the inequality \( x \leq 8.247225 \).
- For \( x = 9.5 \): \( 9.5 \not\leq 8.247225 \) (false)
- For \( x = 8 \): \( 8 \leq 8.247225 \) (true)
- For \( x = 10 \): \( 10 \not\leq 8.247225 \) (false)
- For \( x = 9 \): \( 9 \not\leq 8.247225 \) (false)
The only value that makes the inequality \( \sqrt{4x + 1} \leq 5.83 \) true is:
\[ \boxed{8} \]
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