To determine which values make \( \sqrt{x} + 1 \) a whole number, we need \( \sqrt{x} \) to be a whole number itself. This means \( x \) must be a perfect square.
Let's check each value of \( x \):
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For \( x = 5 \): \[ \sqrt{5} \approx 2.236 \quad \Rightarrow \quad \sqrt{5} + 1 \approx 3.236 \quad \text{(not a whole number)} \]
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For \( x = 24 \): \[ \sqrt{24} \approx 4.899 \quad \Rightarrow \quad \sqrt{24} + 1 \approx 5.899 \quad \text{(not a whole number)} \]
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For \( x = 15 \): \[ \sqrt{15} \approx 3.873 \quad \Rightarrow \quad \sqrt{15} + 1 \approx 4.873 \quad \text{(not a whole number)} \]
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For \( x = -10 \): \[ \sqrt{-10} \quad \text{(not defined in the set of real numbers)} \]
None of the values of \( x = 5, 24, 15, \) or \( -10 \) make \( \sqrt{x} + 1 \) a whole number.
To make \( \sqrt{x} + 1 \) a whole number, \( x \) must be a perfect square (e.g., \( 0, 1, 4, 9, 16, 25, \) etc.).
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