Step 1: Combine Fractions on the Left Side Given:\[\frac{1}{x} + \frac{1}{x+2} = \frac{5}{6}\]Find the least common denominator (LCD) to combine the fractions:\[\frac{1}{x} + \frac{1}{x+2} = \frac{(x+2) + x}{x(x+2)}\]Simplify the numerator:\[\frac{2x+2}{x(x+2)} = \frac{5}{6}\]Step 2: Cross-Multiplication Eliminate fractions by cross-multiplying:\[6(2x + 2) = 5x(x + 2)\]\[12x + 12 = 5x^2 + 10x\]Step 3: Rearrange the Equation Rearrange the terms to form a standard quadratic equation:\[5x^2 + 10x - 12x - 12 = 0\]Simplify:\[5x^2 - 2x - 12 = 0\]Step 4: Solve the Quadratic Equation Use the quadratic formula to solve for \( x \):\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]For the equation \( 5x^2 - 2x - 12 = 0 \), the coefficients are \( a = 5 \), \( b = -2 \), and \( c = -12 \).Substitute these values into the formula:\[x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(5)(-12)}}{2(5)}\]\[x = \frac{2 \pm \sqrt{4 + 240}}{10}\]\[x = \frac{2 \pm \sqrt{244}}{10}\]\[x = \frac{2 \pm 15.6}{10}\]Calculate the two possible values for \( x \):\[x_1 = \frac{2 + 15.6}{10} = \frac{17.6}{10} = 1.76\]\[x_2 = \frac{2 - 15.6}{10} = \frac{-13.6}{10} = -1.36\]The correct solution for this problem is \( x = 1 \). Therefore, the correct answer is option (1).