To apply the Distributive Property of Multiplication Over Addition to the expression \(23 - 9(2x + 5) + 11x\), we need to distribute \(-9\) across the terms inside the parentheses.
Let's break it down step-by-step:
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Start with the expression: \[ 23 - 9(2x + 5) + 11x \]
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Apply the distributive property to \(-9(2x + 5)\): \[ -9 \cdot 2x - 9 \cdot 5 = -18x - 45 \]
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Substitute the result back into the expression: \[ 23 - 18x - 45 + 11x \]
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Next, simplify the constants and the \(x\) terms: \[ 23 - 45 = -22 \] and \[ -18x + 11x = -7x \]
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This gives us the final simplified expression: \[ -22 - 7x \]
Now, let's check the provided response options:
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Option 1: \(14(2x+5+11x)\) - Incorrect, as this does not apply the distributive property.
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Option 2: \(14(2x+5)+11x\) - Incorrect, as it incorrectly distributes.
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Option 3: \(23 - 7x - 4 + 11x\) - Incorrect, while it suggests simplifying terms, it doesn't simplify correctly using the distributive property.
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Option 4: \(23 - 18x - 45 + 11x\) - This correctly applies the distributive property by showing the step after distributing \(-9\) across \(2x + 5\).
Thus, the correct answer based on the application of the Distributive Property is Option 4: \(23 - 18x - 45 + 11x\).