Simplify the expression: 2x + x × (x − x⁻¹)2 - x⁻¹ / (4x⁻¹ − 5 + x)

Question

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Step 1: Rewrite the terms with negative exponents as fractions. Recall that x⁻¹ = 1/x. Substitute this into the given expression: (4x⁻¹ − 5 + x)/(x⁻¹ − 1) = (4/x − 5 + x)/(1/x − 1) Step 2: Find a common denominator for the terms in the numerator and the denominator separately. For the numerator: 4/x − 5 + x = 4/x − 5x/x + (x²)/(x) = (4 − 5x + x²)/(x) For the denominator: 1/x − 1 = 1/x − x/x = (1 − x)/(x) Step 3: Substitute these simplified expressions back into the fraction. ((x² − 5x + 4)/(x))/((1 − x)/(x)) Step 4: Multiply the numerator by the reciprocal of the denominator. (x² − 5x + 4)/(x) × (x)/(1 − x) Step 5: Cancel out the common term x (assuming x ≠ 0). (x² − 5x + 4)/(1 − x) Step 6: Factor the quadratic expression in the numerator. We need two numbers that multiply to 4 and add to −5. These numbers are −1 and −4. So, x² − 5x + 4 = (x − 1)(x − 4). Step 7: Substitute the factored numerator back into the expression. ((x − 1)(x − 4))/(1 − x) Notice that 1 − x is the negative of x − 1, i.e., 1 − x = −(x − 1). Step 8: Substitute −(x − 1) for 1 − x and simplify. ((x − 1)(x − 4))/(−(x − 1)) Cancel the common term (x − 1) (assuming x − 1 ≠ 0): (x − 4)/(−1) = −(x − 4) = −x + 4 The simplified expression is: 4 − x

Simplify the expression: 2x + x * (x - x⁻¹)2 - x⁻¹ / (4x⁻¹ - 5 + x)

Answer

Step-by-step solution

Rewrite the terms with negative exponents as fractions.

Find a common denominator for the terms in the numerator and the denominator separately.

Substitute these simplified expressions back into the fraction.

Multiply the numerator by the reciprocal of the denominator.

Cancel out the common term $x$ (assuming $x \neq 0$ ).

Factor the quadratic expression in the numerator.

Substitute the factored numerator back into the expression.

Substitute $-(x - 1)$ for $1 - x$ and simplify.

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