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Simplifying 2ln(x + 1) + -1ln(3x + -1) = ln(x + -1) Reorder the terms: 2ln(1 + x) + -1ln(3x + -1) = ln(x + -1) (1 * 2ln + x * 2ln) + -1ln(3x + -1) = ln(x + -1) (2ln + 2lnx) + -1ln(3x + -1) = ln(x + -1) Reorder the terms: 2ln + 2lnx + -1ln(-1 + 3x) = ln(x + -1) 2ln + 2lnx + (-1 * -1ln + 3x * -1ln) = ln(x + -1) 2ln + 2lnx + (1ln + -3lnx) = ln(x + -1) Reorder the terms: 2ln + 1ln + 2lnx + -3lnx = ln(x + -1) Combine like terms: 2ln + 1ln = 3ln 3ln + 2lnx + -3lnx = ln(x + -1) Combine like terms: 2lnx + -3lnx = -1lnx 3ln + -1lnx = ln(x + -1) Reorder the terms: 3ln + -1lnx = ln(-1 + x) 3ln + -1lnx = (-1 * ln + x * ln) 3ln + -1lnx = (-1ln + lnx) Solving 3ln + -1lnx = -1ln + lnx Solving for variable 'l'. Move all terms containing l to the left, all other terms to the right. Add 'ln' to each side of the equation. 3ln + ln + -1lnx = -1ln + ln + lnx Combine like terms: 3ln + ln = 4ln 4ln + -1lnx = -1ln + ln + lnx Combine like terms: -1ln + ln = 0 4ln + -1lnx = 0 + lnx 4ln + -1lnx = lnx Add '-1lnx' to each side of the equation. 4ln + -1lnx + -1lnx = lnx + -1lnx Combine like terms: -1lnx + -1lnx = -2lnx 4ln + -2lnx = lnx + -1lnx Combine like terms: lnx + -1lnx = 0 4ln + -2lnx = 0 Factor out the Greatest Common Factor (GCF), '2ln'. 2ln(2 + -1x) = 0 Ignore the factor 2.Subproblem 1
Set the factor 'ln' equal to zero and attempt to solve: Simplifying ln = 0 Solving ln = 0 Move all terms containing l to the left, all other terms to the right. Simplifying ln = 0 The solution to this equation could not be determined. This subproblem is being ignored because a solution could not be determined.Subproblem 2
Set the factor '(2 + -1x)' equal to zero and attempt to solve: Simplifying 2 + -1x = 0 Solving 2 + -1x = 0 Move all terms containing l to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + -1x = 0 + -2 Combine like terms: 2 + -2 = 0 0 + -1x = 0 + -2 -1x = 0 + -2 Combine like terms: 0 + -2 = -2 -1x = -2 Add 'x' to each side of the equation. -1x + x = -2 + x Combine like terms: -1x + x = 0 0 = -2 + x Simplifying 0 = -2 + x The solution to this equation could not be determined. This subproblem is being ignored because a solution could not be determined. The solution to this equation could not be determined.
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