To solve the equation
\[ \frac{1}{x^2} - x + \frac{1}{x} = \frac{5}{x^2} - x, \]
we first identify the common denominator. The common denominator for the fractions involved is \(x^2\).
Now, let's multiply every term in the equation by \(x^2\) to eliminate the fractions:
\[ x^2 \left(\frac{1}{x^2}\right) - x^2(x) + x^2 \left(\frac{1}{x}\right) = x^2 \left(\frac{5}{x^2}\right) - x^2(x). \]
This gives us:
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The first term: \[ x^2 \cdot \frac{1}{x^2} = 1, \]
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The second term: \[ -x^2 \cdot x = -x^3, \]
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The third term: \[ x^2 \cdot \frac{1}{x} = x, \]
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The fourth term (on the right): \[ x^2 \cdot \frac{5}{x^2} = 5, \]
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The fifth term (on the right, same as the second term): \[ -x^2 \cdot x = -x^3. \]
Putting it all together, the equation becomes:
\[ 1 - x^3 + x = 5 - x^3. \]
Now we simplify:
- We observe that the \(-x^3\) terms on both sides cancel out:
The equation simplifies to:
\[ 1 + x = 5. \]
To isolate \(x\), we subtract 1 from both sides:
\[ x = 5 - 1, \] \[ x = 4. \]
Thus, the equivalent expression after multiplying by the common denominator has been demonstrated, and we solved for \(x\). The final answer is \(x = 4\).