To identify which option corrects the error in the given equation \( \frac{1}{2}(2x + 6) + 4(x + 5) = 5x - 17 \), we can start by simplifying both sides.
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Simplify the left-hand side:
- Distribute \( \frac{1}{2} \) to \( (2x + 6) \): \[ \frac{1}{2}(2x + 6) = x + 3 \]
- Distribute \( 4 \) to \( (x + 5) \): \[ 4(x + 5) = 4x + 20 \]
- Combine these two results: \[ x + 3 + 4x + 20 = 5x + 23 \]
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Setting the simplified left side equal to the right side: \[ 5x + 23 = 5x - 17 \]
Now we see what's happening when we compare both sides.
Subtract \( 5x \) from both sides: \[ 23 = -17 \]
Clearly, this does not hold true, indicating an inconsistency which stems from the operations in our original equation.
Now, let's evaluate the response options to see which one could correct the error.
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Option 1: Change \( \frac{1}{2}(2x + 6) \) to \( \frac{1}{2}(2x - 6) \)
- This results in: \[ x - 3 + 4x + 20 = 5x + 17 \]
- This equation becomes \( 5x + 17 = 5x - 17 \), still inconsistent.
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Option 2: Change \( 5x - 17 \) to \( 5x + 17 \)
- The left-hand side would equal \( 5x + 23 = 5x + 17 \), inconsistent, still wouldn’t hold.
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Option 3: Change \( (x + 5) \) to \( (x - 5) \)
- Then it results in: \[ x + 3 + 4(x - 5) = 5x - 17 \] which simplifies to \( 5x - 17 = 5x - 17 \), which holds true. So, this correction works.
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Option 4: Change \( +4(x + 5) \) to \( -4(x + 5) \)
- This gives: \[ x + 3 - 4(x + 5) = 5x - 17 \] resulting in a new total that also ends up being inconsistent.
Only Option 3: Change \( (x + 5) \) to \( (x - 5) \) correctly resolves the equation. Thus:
The correct answer is: Change \( (x + 5) \) to \( (x - 5) \).
1 answer