Algebra Examples

${(\frac{4}{6})}^{x} \geq \frac{36}{16}$

Step 1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln ({(\frac{4}{6})}^{x}) \geq \ln (\frac{36}{16})$

Step 2

Expand the left side.

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Step 2.1

Expand $\ln ({(\frac{4}{6})}^{x})$ by moving $x$ outside the logarithm.

$x \ln (\frac{4}{6}) \geq \ln (\frac{36}{16})$

Step 2.2

Reduce the expression $\frac{4}{6}$ by cancelling the common factors.

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Step 2.2.1

Factor $2$ out of $4$ .

$x \ln (\frac{2 (2)}{6}) \geq \ln (\frac{36}{16})$

Step 2.2.2

Factor $2$ out of $6$ .

$x \ln (\frac{2 \cdot 2}{2 \cdot 3}) \geq \ln (\frac{36}{16})$

Step 2.2.3

Cancel the common factor.

$x \ln (\frac{2 \cdot 2}{2 \cdot 3}) \geq \ln (\frac{36}{16})$

Step 2.2.4

Rewrite the expression.

$x \ln (\frac{2}{3}) \geq \ln (\frac{36}{16})$

Step 2.3

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

$x (\ln (2) - \ln (3)) \geq \ln (\frac{36}{16})$

Step 3

Simplify the left side.

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Step 3.1

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

$x \ln (\frac{2}{3}) \geq \ln (\frac{36}{16})$

Step 4

Simplify the right side.

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Step 4.1

Cancel the common factor of $36$ and $16$ .

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Step 4.1.1

Factor $4$ out of $36$ .

$x \ln (\frac{2}{3}) \geq \ln (\frac{4 (9)}{16})$

Step 4.1.2

Cancel the common factors.

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Step 4.1.2.1

Factor $4$ out of $16$ .

$x \ln (\frac{2}{3}) \geq \ln (\frac{4 \cdot 9}{4 \cdot 4})$

Step 4.1.2.2

Cancel the common factor.

$x \ln (\frac{2}{3}) \geq \ln (\frac{4 \cdot 9}{4 \cdot 4})$

Step 4.1.2.3

Rewrite the expression.

$x \ln (\frac{2}{3}) \geq \ln (\frac{9}{4})$

Step 5

Divide each term in $x \ln (\frac{2}{3}) \geq \ln (\frac{9}{4})$ by $\ln (\frac{2}{3})$ and simplify.

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Step 5.1

Divide each term in $x \ln (\frac{2}{3}) \geq \ln (\frac{9}{4})$ by $\ln (\frac{2}{3})$ .

$\frac{x \ln (\frac{2}{3})}{\ln (\frac{2}{3})} \geq \frac{\ln (\frac{9}{4})}{\ln (\frac{2}{3})}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $\ln (\frac{2}{3})$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{x \ln (\frac{2}{3})}{\ln (\frac{2}{3})} \geq \frac{\ln (\frac{9}{4})}{\ln (\frac{2}{3})}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{\ln (\frac{9}{4})}{\ln (\frac{2}{3})}$

Step 6

The solution consists of all of the true intervals.

$x \leq \frac{\ln (\frac{9}{4})}{\ln (\frac{2}{3})}$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$x \leq \frac{\ln (\frac{9}{4})}{\ln (\frac{2}{3})}$

Interval Notation:

$(- \infty, \frac{\ln (\frac{9}{4})}{\ln (\frac{2}{3})}]$

Step 8