To determine which values for \( x \) satisfy the inequality \( \sqrt{4x + 1} \leq 5.83 \), we can start by squaring both sides of the inequality:
\[ \sqrt{4x + 1} \leq 5.83 \]
Square both sides:
\[ 4x + 1 \leq (5.83)^2 \]
Calculate \( (5.83)^2 \):
\[ (5.83)^2 \approx 33.9889 \]
Now, the inequality becomes:
\[ 4x + 1 \leq 33.9889 \]
Subtract 1 from both sides:
\[ 4x \leq 32.9889 \]
Now divide by 4:
\[ x \leq \frac{32.9889}{4} \approx 8.247225 \]
Now, let's check the provided values (10, 9.5, 9, 8):
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For \( x = 10 \): \[ 10 > 8.247225 \quad \text{(not true)} \]
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For \( x = 9.5 \): \[ 9.5 > 8.247225 \quad \text{(not true)} \]
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For \( x = 9 \): \[ 9 > 8.247225 \quad \text{(not true)} \]
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For \( x = 8 \): \[ 8 \leq 8.247225 \quad \text{(true)} \]
Thus, the only value of \( x \) that makes the inequality \( \sqrt{4x + 1} \leq 5.83 \) true is 8.