The equation $x- \dfrac{7}{x-3}= 3 - \dfrac{7}{x-3} $ has:
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Infinitely many integral roots
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No roots
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One integral root
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Two equal integral roots
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Two equal non-integral roots
The equation simplifies to x = 3 by canceling the identical term -7/(x-3) from both sides. However, if x = 3, the term 7/(x-3) is undefined (division by zero). Therefore, there is no valid solution for x.
Begin by canceling the identical fractional term, negative 7 over (x minus 3), from both sides of the equation, which leaves x equal to 3. Before finalizing the answer, check this solution against the domain restrictions of the original equation, noting that the denominator (x minus 3) becomes zero when x is 3. Since division by zero is undefined, x equals 3 is an extraneous root and must be rejected. Because there are no other possible values for x, the equation has no roots.