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Simplifying ln(x + 31) + -1ln(3 + -2x) = 5ln * 2 Reorder the terms: ln(31 + x) + -1ln(3 + -2x) = 5ln * 2 (31 * ln + x * ln) + -1ln(3 + -2x) = 5ln * 2 (31ln + lnx) + -1ln(3 + -2x) = 5ln * 2 31ln + lnx + (3 * -1ln + -2x * -1ln) = 5ln * 2 31ln + lnx + (-3ln + 2lnx) = 5ln * 2 Reorder the terms: 31ln + -3ln + lnx + 2lnx = 5ln * 2 Combine like terms: 31ln + -3ln = 28ln 28ln + lnx + 2lnx = 5ln * 2 Combine like terms: lnx + 2lnx = 3lnx 28ln + 3lnx = 5ln * 2 Reorder the terms for easier multiplication: 28ln + 3lnx = 5 * 2ln Multiply 5 * 2 28ln + 3lnx = 10ln Solving 28ln + 3lnx = 10ln Solving for variable 'l'. Move all terms containing l to the left, all other terms to the right. Add '-10ln' to each side of the equation. 28ln + -10ln + 3lnx = 10ln + -10ln Combine like terms: 28ln + -10ln = 18ln 18ln + 3lnx = 10ln + -10ln Combine like terms: 10ln + -10ln = 0 18ln + 3lnx = 0 Factor out the Greatest Common Factor (GCF), '3ln'. 3ln(6 + x) = 0 Ignore the factor 3.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 '(6 + x)' equal to zero and attempt to solve: Simplifying 6 + x = 0 Solving 6 + x = 0 Move all terms containing l to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + x = 0 + -6 Combine like terms: 6 + -6 = 0 0 + x = 0 + -6 x = 0 + -6 Combine like terms: 0 + -6 = -6 x = -6 Add '-1x' to each side of the equation. x + -1x = -6 + -1x Combine like terms: x + -1x = 0 0 = -6 + -1x Simplifying 0 = -6 + -1x 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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