Let's analyze Emory's steps one by one to identify where the error occurred in solving the equation \( x + 7 = 3x + (-5) \).
Starting with the original equation:
1. \( x + 7 = 3x - 5 \)
Step 1: (Emory's Step)
Emory stated that he added 1 negative x-tile to both sides. Instead, he should have added 3 negative x-tiles to isolate \( x \).
Correct operation: \[ x - 3x + 7 = -5 \] This simplifies to: \[-2x + 7 = -5\]
Step 2: (Emory's Step)
He added 7 negative unit tiles to both sides, but he should have added 5 positive unit tiles to both sides.
Correct operation: \[ -2x + 7 + 5 = -5 + 5 \] This simplifies to: \[-2x + 12 = 0\]
Step 3: (Emory's Step)
He says he divided the 12 negative unit tiles among the 2 x-tiles. This step is incorrect because he should have set \( -2x = -12 \) based on rearranging the equation to solve for \( x \). Then he would divide by -2:
So when we start from: \[-2x = -12\] Dividing both sides by -2 gives: \[x = 6\]
Step 4: (Emory's Step)
He concluded that the solution is \( x = -6 \), which is incorrect. The correct solution, upon correctly rearranging the terms and solving, yields \( x = 6\).
Based on this analysis, the errors occurred in multiple steps, but the most significant errors were in steps 1 and 2.
Therefore, the answer can be summarized as follows:
- In step 1, he should have added 3 negative x-tiles to both sides.
- In step 2, he should have added 5 positive unit tiles to both sides.
- In step 4, he should have found the solution \( x = 6\).
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