Solve for x 10^(4x+1)>100^(x-2)

$10^{4 x + 1} > 100^{x - 2}$

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

$10^{4 x + 1} > 10^{2 (x - 2)}$

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

$4 x + 1 > 2 (x - 2)$

Solve for $x$ .

Simplify $2 (x - 2)$ .

Rewrite.

$4 x + 1 > 0 + 0 + 2 (x - 2)$

Simplify by adding zeros.

$4 x + 1 > 2 (x - 2)$

Apply the distributive property.

$4 x + 1 > 2 x + 2 \cdot - 2$

Multiply $2$ by $- 2$ .

$4 x + 1 > 2 x - 4$

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

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

$4 x + 1 - 2 x > - 4$

Subtract $2 x$ from $4 x$ .

$2 x + 1 > - 4$

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

Subtract $1$ from both sides of the inequality.

$2 x > - 4 - 1$

Subtract $1$ from $- 4$ .

$2 x > - 5$

Divide each term in $2 x > - 5$ by $2$ and simplify.

Divide each term in $2 x > - 5$ by $2$ .

$\frac{2 x}{2} > \frac{- 5}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} > \frac{- 5}{2}$

Divide $x$ by $1$ .

$x > \frac{- 5}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x > - \frac{5}{2}$

The result can be shown in multiple forms.

Inequality Form:

$x > - \frac{5}{2}$

Interval Notation:

$(- \frac{5}{2}, \infty)$