To determine which equation has no solutions, let's analyze each option:
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−6x + 4 = −6x + 3: If we simplify this, we can add \(6x\) to both sides: \(4 = 3\). This is a false statement, so this equation has no solutions.
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6x + 4 = 6x + 4: This simplifies to \(6x + 4 - 6x - 4 = 0\) or \(0 = 0\), which is a true statement for all \(x\). Therefore, this equation has infinite solutions.
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−6x + 4 = 6x + 3: If we combine like terms, we can add \(6x\) to both sides: \(4 - 3 = 6x + 6x\) leads to \(1 = 12x\), or \(x = \frac{1}{12}\). This equation has one solution.
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6x + 3 = −6x + 3: If we add \(6x\) to both sides, we get \(6x + 6x + 3 = 3\) which simplifies to \(12x + 3 = 3\). Subtracting 3 from both sides gives us \(12x = 0\) or \(x = 0\). This equation has one solution.
Based on this analysis, the equation that has no solutions is:
−6x + 4 = −6x + 3.