If the sum of all the solutions of the equation ⁻¹ ( 1-x 1+x ) + ⁻¹ ( 1+x 1-x ) = 5 6 , where x > -1 and x 1 , is of the form k 3 , then the value of k is _____ .

Step-by-step solution

Given equation: ⁻¹ ( 1-x 1+x ) + ⁻¹ ( 1+x 1-x ) = 5 6 for x > -1, x 1 . We know that for x > -1 , ⁻¹ ( 1-x 1+x ) = 4 - ⁻¹x . The argument of ⁻¹ is 1+x 1-x . Since x > -1 , the numerator 1+x is always positive. The sign of the argument depends on 1-x . Case 1: Let x (-1, 1) . Here, 1-x > 0 , so 1+x 1-x > 0 . Thus, ⁻¹ ( 1+x 1-x ) = ⁻¹ ( 1-x 1+x ) . The equation becomes: 2 ⁻¹ ( 1-x 1+x ) = 5 6 ⁻¹ ( 1-x 1+x ) = 5 12 4 - ⁻¹x = 5 12 ⁻¹x = 4 - 5 12 = - 6 x = (- 6 ) = - 1 3 Since - 1 3 (-1, 1) , this is a valid solution. C