Question

Решить неравенство (1/27)^-2x < 9^3

Answer 1

$(\frac{1}{27})^{-2x}$

Answer 2

To solve the inequality $(\frac{1}{27})^{-2x} < 9^3$, we can follow these steps:

Step 1: Simplify the expression on both sides of the inequality. Step 2: Convert both sides to the same base for easier comparison. Step 3: Solve the simplified inequality for $x$.

Let's proceed with the solution:

Step 1: Simplify the expression on both sides of the inequality.

Recall that $(a^m)^n = a^{mn}$. Applying this property to the left side:

$(\frac{1}{27})^{-2x} = \left(\frac{1}{27}\right)^{-2x} = (27)^{2x} = 27^{2x}$.

Step 2: Convert both sides to the same base.

Since $9 = 3^2$, we can rewrite the right side of the inequality as $9^3 = (3^2)^3 = 3^{2\times 3} = 3^6$.

Now our inequality becomes $27^{2x} < 3^6$.

Step 3: Solve the simplified inequality for $x$.

Now that both sides have the same base, we can compare the exponents:

$2x < 6$.

Next, isolate $x$ by dividing both sides by 2:

$x < \frac{6}{2}$.

$x < 3$.

So, the solution for the inequality is $x < 3$.