Algebra Examples

$\frac{1^{2 x - 10}}{6} \leq \frac{1^{3 x + 13}}{36}$

Step 1

Take the log of both sides of the inequality.

$\ln (\frac{1^{2 x - 10}}{6}) \leq \ln (\frac{1^{3 x + 13}}{36})$

Step 2

Rewrite $\ln (\frac{1^{2 x - 10}}{6})$ as $\ln (1^{2 x - 10}) - \ln (6)$ .

$\ln (1^{2 x - 10}) - \ln (6) \leq \ln (\frac{1^{3 x + 13}}{36})$

Step 3

Expand $\ln (1^{2 x - 10})$ by moving $2 x - 10$ outside the logarithm.

$(2 x - 10) \ln (1) - \ln (6) \leq \ln (\frac{1^{3 x + 13}}{36})$

Step 4

The natural logarithm of $1$ is $0$ .

$(2 x - 10) \cdot 0 - \ln (6) \leq \ln (\frac{1^{3 x + 13}}{36})$

Step 5

Rewrite $\ln (6)$ as $\ln (2 (3))$ .

$(2 x - 10) \cdot 0 - \ln (2 (3)) \leq \ln (\frac{1^{3 x + 13}}{36})$

Step 6

Rewrite $\ln (2 (3))$ as $\ln (2) + \ln (3)$ .

$(2 x - 10) \cdot 0 - (\ln (2) + \ln (3)) \leq \ln (\frac{1^{3 x + 13}}{36})$

Step 7

Apply the distributive property.

$(2 x - 10) \cdot 0 - \ln (2) - \ln (3) \leq \ln (\frac{1^{3 x + 13}}{36})$

Step 8

Multiply $2 x - 10$ by $0$ .

$0 - \ln (2) - \ln (3) \leq \ln (\frac{1^{3 x + 13}}{36})$

Step 9

Subtract $\ln (2)$ from $0$ .

$- \ln (2) - \ln (3) \leq \ln (\frac{1^{3 x + 13}}{36})$

Step 10

Rewrite $\ln (\frac{1^{3 x + 13}}{36})$ as $\ln (1^{3 x + 13}) - \ln (36)$ .

$- \ln (2) - \ln (3) \leq \ln (1^{3 x + 13}) - \ln (36)$

Step 11

Expand $\ln (1^{3 x + 13})$ by moving $3 x + 13$ outside the logarithm.

$- \ln (2) - \ln (3) \leq (3 x + 13) \ln (1) - \ln (36)$

Step 12

The natural logarithm of $1$ is $0$ .

$- \ln (2) - \ln (3) \leq (3 x + 13) \cdot 0 - \ln (36)$

Step 13

Rewrite $\ln (36)$ as $\ln (2^{2} \cdot 3^{2})$ .

$- \ln (2) - \ln (3) \leq (3 x + 13) \cdot 0 - \ln (2^{2} \cdot 3^{2})$

Step 14

Rewrite $\ln (2^{2} \cdot 3^{2})$ as $\ln (2^{2}) + \ln (3^{2})$ .

$- \ln (2) - \ln (3) \leq (3 x + 13) \cdot 0 - (\ln (2^{2}) + \ln (3^{2}))$

Step 15

Simplify each term.

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Step 15.1

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

$- \ln (2) - \ln (3) \leq (3 x + 13) \cdot 0 - (2 \ln (2) + \ln (3^{2}))$

Step 15.2

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

$- \ln (2) - \ln (3) \leq (3 x + 13) \cdot 0 - (2 \ln (2) + 2 \ln (3))$

Step 16

Apply the distributive property.

$- \ln (2) - \ln (3) \leq (3 x + 13) \cdot 0 - (2 \ln (2)) - (2 \ln (3))$

Step 17

Multiply $2$ by $- 1$ .

$- \ln (2) - \ln (3) \leq (3 x + 13) \cdot 0 - 2 \ln (2) - (2 \ln (3))$

Step 18

Multiply $2$ by $- 1$ .

$- \ln (2) - \ln (3) \leq (3 x + 13) \cdot 0 - 2 \ln (2) - 2 \ln (3)$

Step 19

Multiply $3 x + 13$ by $0$ .

$- \ln (2) - \ln (3) \leq 0 - 2 \ln (2) - 2 \ln (3)$

Step 20

Subtract $2 \ln (2)$ from $0$ .

$- \ln (2) - \ln (3) \leq - 2 \ln (2) - 2 \ln (3)$

Step 21

Solve the inequality for $x$ .

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Step 21.1

Simplify each term.

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Step 21.1.1

Simplify $- 2 \log_{2.71828182} (2)$ by moving $2$ inside the logarithm.

$- \log_{2.71828182} (2) - \log_{2.71828182} (3) \leq - \log_{2.71828182} (2^{2}) - 2 \log_{2.71828182} (3)$

Step 21.1.2

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

$- \log_{2.71828182} (2) - \log_{2.71828182} (3) \leq - \log_{2.71828182} (4) - 2 \log_{2.71828182} (3)$

Step 21.1.3

Simplify $- 2 \log_{2.71828182} (3)$ by moving $2$ inside the logarithm.

$- \log_{2.71828182} (2) - \log_{2.71828182} (3) \leq - \log_{2.71828182} (4) - \log_{2.71828182} (3^{2})$

Step 21.1.4

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

$- \log_{2.71828182} (2) - \log_{2.71828182} (3) \leq - \log_{2.71828182} (4) - \log_{2.71828182} (9)$

Step 21.2

Since $- \log_{2.71828182} (2) - \log_{2.71828182} (3) \neq - \log_{2.71828182} (4) - \log_{2.71828182} (9)$ , there are no solutions.

No solution