Step-by-Step Solution

Solve the equation: $\frac{x}{x+1} + \frac{x+1}{x} = \frac{5}{2} \quad (x \ne -1, 0)$

Step-by-Step Solution

Step 1: Substitute reciprocal variable

Let $y = \dfrac{x}{x+1}$. Then $\dfrac{1}{y} = \dfrac{x+1}{x}$: $y + \frac{1}{y} = \frac{5}{2}$

Multiply by $2y$: $2y^2 + 2 = 5y \implies 2y^2 - 5y + 2 = 0$

Step 2: Factorize for $y$

$2y^2 - 4y - y + 2 = 0$ $2y(y - 2) - 1(y - 2) = 0$ $(2y - 1)(y - 2) = 0$

$y = \frac{1}{2} \quad \text{or} \quad y = 2$

Step 3: Back-substitute to find $x$

• Case 1 ($y = \frac{1}{2}$): $\frac{x}{x+1} = \frac{1}{2} \implies 2x = x + 1 \implies x = 1$

• Case 2 ($y = 2$): $\frac{x}{x+1} = 2 \implies x = 2(x + 1) \implies x = 2x + 2 \implies x = -2$

Step 4: Check solutions

• For $x = 1$: $\dfrac{1}{2} + \dfrac{2}{1} = \dfrac{5}{2}$ $\checkmark$
• For $x = -2$: $\dfrac{-2}{-1} + \dfrac{-1}{-2} = 2 + \dfrac{1}{2} = \dfrac{5}{2}$ $\checkmark$

$\boxed{\text{Solution Set: } \{-2, \; 1\}}$