To solve the equation \( x + 1 = -7 \), we need to isolate \( x \).
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Start with the equation: \[ x + 1 = -7 \]
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Subtract 1 from both sides: \[ x = -7 - 1 \]
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Simplify the right side: \[ x = -8 \]
Thus, the integer that makes the addition sentence true is \( \boxed{-8} \).