Solve log_( 1/3 ) (x)> log_(x) (3)- 5/2

Evaluate

lo g_{\frac{1}{3}} (x) > lo g_{x} (3) - \frac{5}{2}

Find the domain

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lo g_{\frac{1}{3}} (x) > lo g_{x} (3) - \frac{5}{2}, x \in (0, 1) \cup (1, + \infty)

Add or subtract both sides

lo g_{\frac{1}{3}} (x) - (lo g_{x} (3) - \frac{5}{2}) > 0

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

lo g_{\frac{1}{3}} (x) - lo g_{x} (3) + \frac{5}{2} > 0

Calculate

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lo g_{\frac{1}{3}} (x) + \frac{1}{lo g _{\frac{1}{3}} ( x )} + \frac{5}{2} > 0

Solve the equation using substitution t = lo g_{\frac{1}{3}} (x)

t + \frac{1}{t} + \frac{5}{2} > 0

Convert the expressions

\frac{2 t ^{2} + 2 + 5 t}{2 t} > 0

Separate the inequality into 2 possible cases

{2 t^{2} + 2 + 5 t > 0 2 t > 0 {2 t^{2} + 2 + 5 t < 0 2 t < 0

Solve the inequality

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{t \in (- \infty, - 2) \cup (- \frac{1}{2}, + \infty) 2 t > 0 {2 t^{2} + 2 + 5 t < 0 2 t < 0

Solve the inequality

{t \in (- \infty, - 2) \cup (- \frac{1}{2}, + \infty) t > 0 {2 t^{2} + 2 + 5 t < 0 2 t < 0

Solve the inequality

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{t \in (- \infty, - 2) \cup (- \frac{1}{2}, + \infty) t > 0 {- 2 < t < - \frac{1}{2} 2 t < 0

Solve the inequality

{t \in (- \infty, - 2) \cup (- \frac{1}{2}, + \infty) t > 0 {- 2 < t < - \frac{1}{2} t < 0

Find the intersection

t > 0 {- 2 < t < - \frac{1}{2} t < 0

Find the intersection

t > 0 - 2 < t < - \frac{1}{2}

Find the union

t \in (- 2, - \frac{1}{2}) \cup (0, + \infty)

Substitute back

- 2 < lo g_{\frac{1}{3}} (x) < - \frac{1}{2} lo g_{\frac{1}{3}} (x) > 0

Solve the inequality for x

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3 < x < 9 lo g_{\frac{1}{3}} (x) > 0

Solve the inequality for x

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3 < x < 9 x < 1

Find the union

x \in (- \infty, 1) \cup (3, 9)

Check if the solution is in the defined range

x \in (- \infty, 1) \cup (3, 9), x \in (0, 1) \cup (1, + \infty)

Solution

x \in (0, 1) \cup (3, 9)