Solve the Inequality for x 3^x<9^(x-2)

$3^{x} < 9^{x - 2}$

Step 1

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

$3^{x} < 3^{2 (x - 2)}$

Step 2

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

$x < 2 (x - 2)$

Step 3

Solve for $x$ .

Simplify $2 (x - 2)$ .

Apply the distributive property.

$x < 2 x + 2 \cdot - 2$

Multiply $2$ by $- 2$ .

$x < 2 x - 4$

Move all terms containing $x$ to the left side of the inequality.

Subtract $2 x$ from both sides of the inequality.

$x - 2 x < - 4$

Subtract $2 x$ from $x$ .

$- x < - 4$

Divide each term in $- x < - 4$ by $- 1$ and simplify.

Divide each term in $- x < - 4$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} > \frac{- 4}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{- 4}{- 1}$

Divide $x$ by $1$ .

$x > \frac{- 4}{- 1}$

Simplify the right side.

Divide $- 4$ by $- 1$ .

$x > 4$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$x > 4$

Interval Notation:

$(4, \infty)$

Step 5

Solve the Inequality for x 3^x<9^(x-2)

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