Solve for x (0.6)^(x-3)>(0.36)^(-x-1)

${(0.6)}^{x - 3} > {(0.36)}^{- x - 1}$

Step 1

Take the log of both sides of the inequality.

$\ln ({0.6}^{x - 3}) > \ln ({0.36}^{- x - 1})$

Step 2

Expand $\ln ({0.6}^{x - 3})$ by moving $x - 3$ outside the logarithm.

$(x - 3) \ln (0.6) > \ln ({0.36}^{- x - 1})$

Step 3

Expand $\ln ({0.36}^{- x - 1})$ by moving $- x - 1$ outside the logarithm.

$(x - 3) \ln (0.6) > (- x - 1) \ln (0.36)$

Step 4

Solve the inequality for $x$ .

Simplify $(x - 3) \ln (0.6)$ .

Rewrite.

$0 + 0 + (x - 3) \ln (0.6) > (- x - 1) \ln (0.36)$

Simplify by adding zeros.

$(x - 3) \ln (0.6) > (- x - 1) \ln (0.36)$

Apply the distributive property.

$x \ln (0.6) - 3 \ln (0.6) > (- x - 1) \ln (0.36)$

Simplify $(- x - 1) \ln (0.36)$ .

Apply the distributive property.

$x \ln (0.6) - 3 \ln (0.6) > - x \ln (0.36) - 1 \ln (0.36)$

Rewrite $- 1 \ln (0.36)$ as $- \ln (0.36)$ .

$x \ln (0.6) - 3 \ln (0.6) > - x \ln (0.36) - \ln (0.36)$

Add $x \ln (0.36)$ to both sides of the inequality.

$x \ln (0.6) - 3 \ln (0.6) + x \ln (0.36) > - \ln (0.36)$

Add $3 \ln (0.6)$ to both sides of the inequality.

$x \ln (0.6) + x \ln (0.36) > - \ln (0.36) + 3 \ln (0.6)$

Factor $x$ out of $x \ln (0.6) + x \ln (0.36)$ .

Factor $x$ out of $x \ln (0.6)$ .

$x (\ln (0.6)) + x \ln (0.36) > - \ln (0.36) + 3 \ln (0.6)$

Factor $x$ out of $x \ln (0.36)$ .

$x (\ln (0.6)) + x (\ln (0.36)) > - \ln (0.36) + 3 \ln (0.6)$

Factor $x$ out of $x (\ln (0.6)) + x (\ln (0.36))$ .

$x (\ln (0.6) + \ln (0.36)) > - \ln (0.36) + 3 \ln (0.6)$

Divide each term in $x (\ln (0.6) + \ln (0.36)) > - \ln (0.36) + 3 \ln (0.6)$ by $\ln (0.6) + \ln (0.36)$ and simplify.

Divide each term in $x (\ln (0.6) + \ln (0.36)) > - \ln (0.36) + 3 \ln (0.6)$ by $\ln (0.6) + \ln (0.36)$ .

$\frac{x (\ln (0.6) + \ln (0.36))}{\ln (0.6) + \ln (0.36)} > \frac{- \ln (0.36)}{\ln (0.6) + \ln (0.36)} + \frac{3 \ln (0.6)}{\ln (0.6) + \ln (0.36)}$

Simplify the left side.

Cancel the common factor of $\ln (0.6) + \ln (0.36)$ .

Cancel the common factor.

$\frac{x (\ln (0.6) + \ln (0.36))}{\ln (0.6) + \ln (0.36)} > \frac{- \ln (0.36)}{\ln (0.6) + \ln (0.36)} + \frac{3 \ln (0.6)}{\ln (0.6) + \ln (0.36)}$

Divide $x$ by $1$ .

$x > \frac{- \ln (0.36)}{\ln (0.6) + \ln (0.36)} + \frac{3 \ln (0.6)}{\ln (0.6) + \ln (0.36)}$

Simplify the right side.

Combine the numerators over the common denominator.

$x > \frac{- \ln (0.36) + 3 \ln (0.6)}{\ln (0.6) + \ln (0.36)}$

Factor $- 1$ out of $- \ln (0.36)$ .

$x > \frac{- \ln (0.36) + 3 \ln (0.6)}{\ln (0.6) + \ln (0.36)}$

Factor $- 1$ out of $3 \ln (0.6)$ .

$x > \frac{- \ln (0.36) - (- 3 \ln (0.6))}{\ln (0.6) + \ln (0.36)}$

Factor $- 1$ out of $- \ln (0.36) - (- 3 \ln (0.6))$ .

$x > \frac{- (\ln (0.36) - 3 \ln (0.6))}{\ln (0.6) + \ln (0.36)}$

Simplify the expression.

Rewrite $- (\ln (0.36) - 3 \ln (0.6))$ as $- 1 (\ln (0.36) - 3 \ln (0.6))$ .

$x > \frac{- 1 (\ln (0.36) - 3 \ln (0.6))}{\ln (0.6) + \ln (0.36)}$

Move the negative in front of the fraction.

$x > - \frac{\ln (0.36) - 3 \ln (0.6)}{\ln (0.6) + \ln (0.36)}$

Step 5

The solution consists of all of the true intervals.

$x < - \frac{\ln (0.36) - 3 \ln (0.6)}{\ln (0.6) + \ln (0.36)}$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$x < - \frac{\ln (0.36) - 3 \ln (0.6)}{\ln (0.6) + \ln (0.36)}$

Interval Notation:

$(- \infty, - \frac{\ln (0.36) - 3 \ln (0.6)}{\ln (0.6) + \ln (0.36)})$

Step 7

Solve for x (0.6)^(x-3)>(0.36)^(-x-1)

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