To rewrite the equation \(\frac{4x}{3x-12} + 3 = 10\) in two equations joined by "AND," we need to isolate the fraction and then set up a corresponding equation that reflects the relationship.
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Start with the original equation: \[ \frac{4x}{3x-12} + 3 = 10 \]
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Subtract 3 from both sides to isolate the fraction: \[ \frac{4x}{3x-12} = 10 - 3 \] \[ \frac{4x}{3x-12} = 7 \]
Now, we have two statements that can be expressed as separate equations:
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The fraction equation: \[ \frac{4x}{3x-12} = 7 \]
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The condition that we simplified (the right side of the original equation): \[ 3x - 12 \neq 0 \] (because the denominator cannot equal zero)
Now, we can combine these into a statement with "AND":
\[ \frac{4x}{3x-12} = 7 \quad \text{AND} \quad 3x - 12 \neq 0 \]
So the final answer is:
\[ \frac{4x}{3x-12} = 7 \quad \text{AND} \quad 3x \neq 12 \]