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Simplifying ln(x + 1) + ln(x + -1) = ln(3x + 5) + lnx Reorder the terms: ln(1 + x) + ln(x + -1) = ln(3x + 5) + lnx (1 * ln + x * ln) + ln(x + -1) = ln(3x + 5) + lnx (1ln + lnx) + ln(x + -1) = ln(3x + 5) + lnx Reorder the terms: 1ln + lnx + ln(-1 + x) = ln(3x + 5) + lnx 1ln + lnx + (-1 * ln + x * ln) = ln(3x + 5) + lnx 1ln + lnx + (-1ln + lnx) = ln(3x + 5) + lnx Reorder the terms: 1ln + -1ln + lnx + lnx = ln(3x + 5) + lnx Combine like terms: 1ln + -1ln = 0 0 + lnx + lnx = ln(3x + 5) + lnx lnx + lnx = ln(3x + 5) + lnx Combine like terms: lnx + lnx = 2lnx 2lnx = ln(3x + 5) + lnx Reorder the terms: 2lnx = ln(5 + 3x) + lnx 2lnx = (5 * ln + 3x * ln) + lnx 2lnx = (5ln + 3lnx) + lnx Combine like terms: 3lnx + lnx = 4lnx 2lnx = 5ln + 4lnx Solving 2lnx = 5ln + 4lnx Solving for variable 'l'. Move all terms containing l to the left, all other terms to the right. Add '-5ln' to each side of the equation. -5ln + 2lnx = 5ln + -5ln + 4lnx Combine like terms: 5ln + -5ln = 0 -5ln + 2lnx = 0 + 4lnx -5ln + 2lnx = 4lnx Add '-4lnx' to each side of the equation. -5ln + 2lnx + -4lnx = 4lnx + -4lnx Combine like terms: 2lnx + -4lnx = -2lnx -5ln + -2lnx = 4lnx + -4lnx Combine like terms: 4lnx + -4lnx = 0 -5ln + -2lnx = 0 Factor out the Greatest Common Factor (GCF), '-1ln'. -1ln(5 + 2x) = 0 Ignore the factor -1.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 '(5 + 2x)' equal to zero and attempt to solve: Simplifying 5 + 2x = 0 Solving 5 + 2x = 0 Move all terms containing l to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + 2x = 0 + -5 Combine like terms: 5 + -5 = 0 0 + 2x = 0 + -5 2x = 0 + -5 Combine like terms: 0 + -5 = -5 2x = -5 Add '-2x' to each side of the equation. 2x + -2x = -5 + -2x Combine like terms: 2x + -2x = 0 0 = -5 + -2x Simplifying 0 = -5 + -2x 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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