G(x)=x-lnx

Solution for G(x)=x-lnx equation:

Simplifying
G(x) = x + -1lnx

Multiply G * x
xG = x + -1lnx

Reorder the terms:
xG = -1lnx + x

Solving
xG = -1lnx + x

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add 'lnx' to each side of the equation.
lnx + xG = -1lnx + lnx + x

Combine like terms: -1lnx + lnx = 0
lnx + xG = 0 + x
lnx + xG = x

Add '-1x' to each side of the equation.
lnx + -1x + xG = x + -1x

Combine like terms: x + -1x = 0
lnx + -1x + xG = 0

Factor out the Greatest Common Factor (GCF), 'x'.
x(ln + -1 + G) = 0

Subproblem 1
Set the factor 'x' equal to zero and attempt to solve:

Simplifying
x = 0

Solving
x = 0

Move all terms containing x to the left, all other terms to the right.

Simplifying
x = 0
Subproblem 2
Set the factor '(ln + -1 + G)' equal to zero and attempt to solve:

Simplifying
ln + -1 + G = 0

Reorder the terms:
-1 + G + ln = 0

Solving
-1 + G + ln = 0

Move all terms containing x to the left, all other terms to the right.

Add '1' to each side of the equation.
-1 + G + 1 + ln = 0 + 1

Reorder the terms:
-1 + 1 + G + ln = 0 + 1

Combine like terms: -1 + 1 = 0
0 + G + ln = 0 + 1
G + ln = 0 + 1

Combine like terms: 0 + 1 = 1
G + ln = 1

Add '-1G' to each side of the equation.
G + -1G + ln = 1 + -1G

Combine like terms: G + -1G = 0
0 + ln = 1 + -1G
ln = 1 + -1G

Add '-1ln' to each side of the equation.
ln + -1ln = 1 + -1G + -1ln

Combine like terms: ln + -1ln = 0
0 = 1 + -1G + -1ln

Simplifying
0 = 1 + -1G + -1ln

The solution to this equation could not be determined.

This subproblem is being ignored because a solution could not be determined.
Solution
x = {0}