# Solve the equations and inequalities: frac{2^{x}}{3}leqfrac{5^{x}}{4}

Question
Logarithms
Solve the equations and inequalities: $$\displaystyle{\frac{{{2}^{{{x}}}}}{{{3}}}}\leq{\frac{{{5}^{{{x}}}}}{{{4}}}}$$

2020-10-28
Take the logarithm of both sides of the inequality to remove the variable from the exponent. Inequality Form: $$\displaystyle{x}≥\frac{{{\ln{{\left({3}\right)}}}−{\ln{{\left({4}\right)}}}}}{{{\ln{{\left({2}\right)}}}−{\ln{{\left({5}\right)}}}}}$$ Interval Notation: $$\displaystyle{\left[\frac{{{\ln{{\left({3}\right)}}}−{\ln{{\left({4}\right)}}}}}{{{\ln{{\left({2}\right)}}}−{\ln{{\left({5}\right)}}}}},∞\right)}$$

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To solve: The system of congruence $$\displaystyle{x}={2}{\left({b}\text{mod}{3}\right)},{x}={1}{\left({b}\text{mod}{4}\right)},\ {\quad\text{and}\quad}\ {x}={3}{\left({b}\text{mod}{5}\right)}$$ using the method of back substitution.
$$|\frac{2}{x}-4|<3$$