The property used in these equations is the addition (or subtraction) property of inequality. This property states that when you add or subtract the same number from both sides of an inequality, the direction of the inequality remains the same.
Let's break it down step by step:
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Starting equation: \[ 1 - 3x - 8x > 3x + 15 \] Using the addition property, we rearrange terms to isolate \(x\).
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Combine like terms: \[ 1 - 11x > 3x + 15 \]
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Subtract \(3x\) from both sides: \[ 1 - 11x - 3x > 15 \] Which simplifies to: \[ 1 - 14x > 15 \]
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Subtract \(1\) from both sides: \[ -14x > 14 \]
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Finally, divide both sides by \(-14\): When dividing or multiplying by a negative number, the direction of the inequality changes: \[ x < -1 \]
Thus, throughout these steps, the key property utilized is properly handling inequalities while applying the addition and multiplication properties, especially when a negative number is involved in the process.