Solve for x 3^(2x-1)>=1/243

$3^{2 x - 1} \geq \frac{1}{243}$

Step 1

Raise $243$ to the power of $1$ .

$3^{2 x - 1} \geq \frac{1}{243^{1}}$

Step 2

Move $243^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$3^{2 x - 1} \geq 243^{- 1}$

Step 3

Create equivalent expressions in the equation that all have equal bases.

$3^{2 x - 1} \geq 3^{- 5}$

Step 4

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$2 x - 1 \geq - 5$

Step 5

Solve for $x$ .

Move all terms not containing $x$ to the right side of the inequality.

Add $1$ to both sides of the inequality.

$2 x \geq - 5 + 1$

Add $- 5$ and $1$ .

$2 x \geq - 4$

Divide each term in $2 x \geq - 4$ by $2$ and simplify.

Divide each term in $2 x \geq - 4$ by $2$ .

$\frac{2 x}{2} \geq \frac{- 4}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{- 4}{2}$

Divide $x$ by $1$ .

$x \geq \frac{- 4}{2}$

Simplify the right side.

Divide $- 4$ by $2$ .

$x \geq - 2$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$x \geq - 2$

Interval Notation:

$[- 2, \infty)$

Step 7