Given info:- 5/(x-3) - 4/(x+4) = 1/x. And here the problem has asked to find out the value of x ?
Let's solve!!
Given equation is 5/(x-3) - 4/(x+4) = 1/x
⇛ [5(x+4) - 4(x-3)]/(x-3)(x+4) = 1/x
⇛ (5x+20-4x+12)/(x-3)(x+4) = 1/x
⇛ (x+32)/(x-3)(x+4) = 1/x
On applying cross multiplication then
⇛ x(x+32) = (x-3)(x+4)
⇛ x²+32x = x(x+4)-3(x+4)
⇛ x²+32x = x²+4x-3x-12
⇛ x²+32x = x²+x-12
⇛ x²+32x-x²-x = -12
⇛ (x²-x²)+(32x-x) = -12
⇛ 0+31x = -12
⇛ 31x = -12
⇛ x = -12/31
∴ x = -12/31
The value of x for the given problem -12/31
VERIFICATION:
Check
If x = -12/31 then LHS of the equation is
5/(x-3) - 4/(x+4)
⇛ 5/[(-12/31)-3] -4/[(-12/31)+4]
⇛ 5/[(-12-93)/31] -4/[(-12+124)/31]
⇛ 5/(-105/31) - 4/(112)/31
⇛ (5×-31)/105 - (4×31)/112
⇛ (-115/105)-(124/112)
⇛ (-31/21) - (31/28)
⇛ (-31)[(1/21)+(1/28)]
⇛ (-31)(21+28)/(21×28)
⇛ (-31×49)/(21×28)
⇛ (-31×7)/(21×4)
⇛-31/12
And RHS = 1/x
⇛ 1/(-12/31)
⇛ -31/12
⇛ LHS = RHS is true for x = -31/12