Pre-Algebra Examples

$\frac{3}{5^{2 x - 3}} > \frac{5}{3^{x + 2}}$

Step 1

Take the log of both sides of the inequality.

$\ln (\frac{3}{5^{2 x - 3}}) > \ln (\frac{5}{3^{x + 2}})$

Step 2

Rewrite $\ln (\frac{3}{5^{2 x - 3}})$ as $\ln (3) - \ln (5^{2 x - 3})$ .

$\ln (3) - \ln (5^{2 x - 3}) > \ln (\frac{5}{3^{x + 2}})$

Step 3

Expand $\ln (5^{2 x - 3})$ by moving $2 x - 3$ outside the logarithm.

$\ln (3) - ((2 x - 3) \ln (5)) > \ln (\frac{5}{3^{x + 2}})$

Step 4

Remove parentheses.

$\ln (3) - (2 x - 3) \ln (5) > \ln (\frac{5}{3^{x + 2}})$

Step 5

Rewrite $\ln (\frac{5}{3^{x + 2}})$ as $\ln (5) - \ln (3^{x + 2})$ .

$\ln (3) - (2 x - 3) \ln (5) > \ln (5) - \ln (3^{x + 2})$

Step 6

Expand $\ln (3^{x + 2})$ by moving $x + 2$ outside the logarithm.

$\ln (3) - (2 x - 3) \ln (5) > \ln (5) - ((x + 2) \ln (3))$

Step 7

Remove parentheses.

$\ln (3) - (2 x - 3) \ln (5) > \ln (5) - (x + 2) \ln (3)$

Step 8

Solve the inequality for $x$ .

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Step 8.1

Simplify the left side.

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Step 8.1.1

Simplify each term.

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Step 8.1.1.1

Apply the distributive property.

$\ln (3) + (- (2 x) + 3) \ln (5) = \ln (5) - (x + 2) \ln (3)$

Step 8.1.1.2

Multiply $2$ by $- 1$ .

$\ln (3) + (- 2 x + 3) \ln (5) = \ln (5) - (x + 2) \ln (3)$

Step 8.1.1.3

Multiply $- 1$ by $- 3$ .

$\ln (3) + (- 2 x + 3) \ln (5) = \ln (5) - (x + 2) \ln (3)$

Step 8.1.1.4

Apply the distributive property.

$\ln (3) - 2 x \ln (5) + 3 \ln (5) = \ln (5) - (x + 2) \ln (3)$

Step 8.1.2

Reorder $\ln (3)$ and $- 2 x \ln (5)$ .

$- 2 x \ln (5) + \ln (3) + 3 \ln (5) = \ln (5) - (x + 2) \ln (3)$

Step 8.2

Simplify the right side.

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Step 8.2.1

Simplify each term.

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Step 8.2.1.1

Apply the distributive property.

$- 2 x \ln (5) + \ln (3) + 3 \ln (5) = \ln (5) + (- x - 1 \cdot 2) \ln (3)$

Step 8.2.1.2

Multiply $- 1$ by $2$ .

$- 2 x \ln (5) + \ln (3) + 3 \ln (5) = \ln (5) + (- x - 2) \ln (3)$

Step 8.2.1.3

Apply the distributive property.

$- 2 x \ln (5) + \ln (3) + 3 \ln (5) = \ln (5) - x \ln (3) - 2 \ln (3)$

Step 8.2.2

Reorder $\ln (5)$ and $- x \ln (3)$ .

$- 2 x \ln (5) + \ln (3) + 3 \ln (5) = - x \ln (3) + \ln (5) - 2 \ln (3)$

Step 8.3

Simplify the left side.

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Step 8.3.1

Simplify $- 2 x \ln (5) + \ln (3) + 3 \ln (5)$ .

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Step 8.3.1.1

Simplify each term.

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Step 8.3.1.1.1

Simplify $- 2 \ln (5)$ by moving $2$ inside the logarithm.

$x (- \ln (5^{2})) + \ln (3) + 3 \ln (5) = - x \ln (3) + \ln (5) - 2 \ln (3)$

Step 8.3.1.1.2

Raise $5$ to the power of $2$ .

$x (- \ln (25)) + \ln (3) + 3 \ln (5) = - x \ln (3) + \ln (5) - 2 \ln (3)$

Step 8.3.1.1.3

Simplify $3 \ln (5)$ by moving $3$ inside the logarithm.

$x (- \ln (25)) + \ln (3) + \ln (5^{3}) = - x \ln (3) + \ln (5) - 2 \ln (3)$

Step 8.3.1.1.4

Raise $5$ to the power of $3$ .

$x (- \ln (25)) + \ln (3) + \ln (125) = - x \ln (3) + \ln (5) - 2 \ln (3)$

Step 8.3.1.2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$x (- \ln (25)) + \ln (3 \cdot 125) = - x \ln (3) + \ln (5) - 2 \ln (3)$

Step 8.3.1.3

Multiply $3$ by $125$ .

$x (- \ln (25)) + \ln (375) = - x \ln (3) + \ln (5) - 2 \ln (3)$

Step 8.3.1.4

Reorder factors in $x (- \ln (25)) + \ln (375)$ .

$- x \ln (25) + \ln (375) = - x \ln (3) + \ln (5) - 2 \ln (3)$

Step 8.4

Simplify the right side.

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Step 8.4.1

Simplify $- x \ln (3) + \ln (5) - 2 \ln (3)$ .

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Step 8.4.1.1

Simplify each term.

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Step 8.4.1.1.1

Simplify $- 2 \ln (3)$ by moving $2$ inside the logarithm.

$- x \ln (25) + \ln (375) = - x \ln (3) + \ln (5) - \ln (3^{2})$

Step 8.4.1.1.2

Raise $3$ to the power of $2$ .

$- x \ln (25) + \ln (375) = - x \ln (3) + \ln (5) - \ln (9)$

Step 8.4.1.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$- x \ln (25) + \ln (375) = - x \ln (3) + \ln (\frac{5}{9})$

Step 8.5

Move all the terms containing a logarithm to the left side of the equation.

$- x \ln (25) + \ln (375) + x \ln (3) - \ln (\frac{5}{9}) = 0$

Step 8.6

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$- x \ln (25) + x \ln (3) + \ln (\frac{375}{\frac{5}{9}}) = 0$

Step 8.7

Simplify each term.

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Step 8.7.1

Multiply the numerator by the reciprocal of the denominator.

$- x \ln (25) + x \ln (3) + \ln (375 (\frac{9}{5})) = 0$

Step 8.7.2

Cancel the common factor of $5$ .

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Step 8.7.2.1

Factor $5$ out of $375$ .

$- x \ln (25) + x \ln (3) + \ln (5 (75) (\frac{9}{5})) = 0$

Step 8.7.2.2

Cancel the common factor.

$- x \ln (25) + x \ln (3) + \ln (5 \cdot (75 (\frac{9}{5}))) = 0$

Step 8.7.2.3

Rewrite the expression.

$- x \ln (25) + x \ln (3) + \ln (75 \cdot 9) = 0$

Step 8.7.3

Multiply $75$ by $9$ .

$- x \ln (25) + x \ln (3) + \ln (675) = 0$

Step 8.8

Subtract $\ln (675)$ from both sides of the equation.

$- x \ln (25) + x \ln (3) = - \ln (675)$

Step 8.9

Factor $x$ out of $- x \ln (25) + x \ln (3)$ .

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Step 8.9.1

Factor $x$ out of $- x \ln (25)$ .

$x (- 1 \ln (25)) + x \ln (3) = - \ln (675)$

Step 8.9.2

Factor $x$ out of $x \ln (3)$ .

$x (- 1 \ln (25)) + x (\ln (3)) = - \ln (675)$

Step 8.9.3

Factor $x$ out of $x (- 1 \ln (25)) + x (\ln (3))$ .

$x (- 1 \ln (25) + \ln (3)) = - \ln (675)$

Step 8.10

Rewrite $- 1 \ln (25)$ as $- \ln (25)$ .

$x (- \ln (25) + \ln (3)) = - \ln (675)$

Step 8.11

Divide each term in $x (- \ln (25) + \ln (3)) = - \ln (675)$ by $- \ln (25) + \ln (3)$ and simplify.

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Step 8.11.1

Divide each term in $x (- \ln (25) + \ln (3)) = - \ln (675)$ by $- \ln (25) + \ln (3)$ .

$\frac{x (- \ln (25) + \ln (3))}{- \ln (25) + \ln (3)} = \frac{- \ln (675)}{- \ln (25) + \ln (3)}$

Step 8.11.2

Simplify the left side.

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Step 8.11.2.1

Cancel the common factor of $- \ln (25) + \ln (3)$ .

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Step 8.11.2.1.1

Cancel the common factor.

$\frac{x (- \ln (25) + \ln (3))}{- \ln (25) + \ln (3)} = \frac{- \ln (675)}{- \ln (25) + \ln (3)}$

Step 8.11.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \ln (675)}{- \ln (25) + \ln (3)}$

Step 8.11.3

Simplify the right side.

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Step 8.11.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (675)}{- \ln (25) + \ln (3)}$

Step 8.11.3.2

Factor $- 1$ out of $- \ln (25)$ .

$x = - \frac{\ln (675)}{- \ln (25) + \ln (3)}$

Step 8.11.3.3

Factor $- 1$ out of $\ln (3)$ .

$x = - \frac{\ln (675)}{- \ln (25) - 1 (- \ln (3))}$

Step 8.11.3.4

Factor $- 1$ out of $- \ln (25) - 1 (- \ln (3))$ .

$x = - \frac{\ln (675)}{- (\ln (25) - \ln (3))}$

Step 8.11.3.5

Simplify the expression.

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Step 8.11.3.5.1

Rewrite $- (\ln (25) - \ln (3))$ as $- 1 (\ln (25) - \ln (3))$ .

$x = - \frac{\ln (675)}{- 1 (\ln (25) - \ln (3))}$

Step 8.11.3.5.2

Move the negative in front of the fraction.

$x = - - \frac{\ln (675)}{\ln (25) - \ln (3)}$

Step 8.11.3.5.3

Multiply $- 1$ by $- 1$ .

$x = 1 \frac{\ln (675)}{\ln (25) - \ln (3)}$

Step 8.11.3.5.4

Multiply $\frac{\ln (675)}{\ln (25) - \ln (3)}$ by $1$ .

$x = \frac{\ln (675)}{\ln (25) - \ln (3)}$

Step 9

The solution consists of all of the true intervals.

$x < \frac{\ln (675)}{\ln (25) - \ln (3)}$

Step 10