To solve the inequality \( \sqrt{m} + 17 < 24 \), we will first isolate \( \sqrt{m} \).
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Subtract 17 from both sides:
\[ \sqrt{m} < 24 - 17 \]
This simplifies to:
\[ \sqrt{m} < 7 \]
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Next, we square both sides to eliminate the square root (keeping in mind that squaring both sides is valid since both sides are positive):
\[ m < 7^2 \]
This gives us:
\[ m < 49 \]
Now, we will test the given values:
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For \( m = 49 \): \[ \sqrt{49} + 17 = 7 + 17 = 24 \quad \text{(not less than 24)} \]
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For \( m = 48 \): \[ \sqrt{48} + 17 \approx 6.93 + 17 \approx 23.93 < 24 \quad \text{(true)} \]
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For \( m = 50 \): \[ \sqrt{50} + 17 \approx 7.07 + 17 \approx 24.07 \quad \text{(not less than 24)} \]
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For \( m = 64 \): \[ \sqrt{64} + 17 = 8 + 17 = 25 \quad \text{(not less than 24)} \]
Based on the tests, the only value for \( m \) that makes the inequality \( \sqrt{m} + 17 < 24 \) true is:
\[ \boxed{48} \]