Question

1/5^4x-17 >=125 решите неравенство

Answer 1

(1/5)^(4х-17)≥125
5^(17-4х)≥5³
17-4х≥3
-4х≥-14
х≤14/4
х≤7/2
х≤3,5

Answer 2

5^(-4x+17) >=5^3

-4x+17>=3

-4x>=-14

4x>=14

x>=14/4

x>=7/2

x>=3,5

[3,5;+oo)

Answer 3

To solve the inequality $\frac{1}{{5^{4x-17}}} \geq 125$, follow these steps:

Step 1: Rewrite 125 as a fraction with a denominator of $5^{4x-17}$. Step 2: Set up the inequality using the common denominator. Step 3: Solve the resulting inequality for $x$.

Let's proceed with the steps:

Step 1: Rewrite 125 as a fraction with a denominator of $5^{4x-17}$: $125 = \frac{125}{1} = \frac{5^3}{1} = \frac{5^{3+4x-17}}{5^{4x-17}} = \frac{5^{4x-14}}{5^{4x-17}}$.

Step 2: Set up the inequality using the common denominator: $\frac{1}{{5^{4x-17}}} \geq \frac{5^{4x-14}}{5^{4x-17}}$.

Step 3: Solve the resulting inequality for $x$: To make the inequality easier to solve, let's multiply both sides by $5^{4x-17}$ (note that $5^{4x-17}$ is positive, so the inequality direction remains the same):

$5^{4x-17} \cdot \frac{1}{{5^{4x-17}}} \geq 5^{4x-17} \cdot \frac{5^{4x-14}}{5^{4x-17}}$.

On the left side, $5^{4x-17} \cdot \frac{1}{{5^{4x-17}}}$ simplifies to 1, and on the right side, $5^{4x-17} \cdot \frac{5^{4x-14}}{5^{4x-17}}$ simplifies to $5^{-14}$:

$1 \geq 5^{-14}$.

Now, $5^{-14}$ is a positive number (approximately $1.6 \times 10^{-11}$), and any positive number is greater than or equal to 1. So, the inequality holds for all real values of $x$.

Thus, the solution to the inequality is $\boxed{(-\infty, \infty)}$, meaning that any real value of $x$ will satisfy the inequality.