Solve the Inequality for x 25^(x+1)<=125^x

$25^{x + 1} \leq 125^{x}$

Step 1

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

$5^{2 (x + 1)} \leq 5^{3 x}$

Step 2

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

$2 (x + 1) \leq 3 x$

Step 3

Solve for $x$ .

Simplify $2 (x + 1)$ .

Start simplifying.

$0 + 0 + 2 (x + 1) \leq 3 x$

Simplify by adding zeros.

$2 (x + 1) \leq 3 x$

Apply the distributive property.

$2 x + 2 \cdot 1 \leq 3 x$

Multiply $2$ by $1$ .

$2 x + 2 \leq 3 x$

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

Subtract $3 x$ from both sides of the inequality.

$2 x + 2 - 3 x \leq 0$

Subtract $3 x$ from $2 x$ .

$- x + 2 \leq 0$

Subtract $2$ from both sides of the inequality.

$- x \leq - 2$

Divide each term in $- x \leq - 2$ by $- 1$ and simplify.

Divide each term in $- x \leq - 2$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{- 2}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{- 2}{- 1}$

Divide $x$ by $1$ .

$x \geq \frac{- 2}{- 1}$

Simplify the right side.

Divide $- 2$ by $- 1$ .

$x \geq 2$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$x \geq 2$

Interval Notation:

$[2, \infty)$

Step 5

Solve the Inequality for x 25^(x+1)<=125^x

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