6. (1 балл) Решите неравенство: (1/5)^4x–6 < 25

Решение:

Представим обе части неравенства в виде степени с основанием \( 1/5 \) или \( 5 \).

Вариант 1: основание \( 1/5 \)

\( 25 = \frac{1}{1/25} = (1/5)^2 \)

Неравенство принимает вид:

\[ (1/5)^{4x-6} < (1/5)^2 \]

Так как основание степени \( 1/5 < 1 \), при раскрытии неравенства меняем знак на противоположный:

\[ 4x - 6 > 2 \]\[ 4x > 8 \]\[ x > 2 \]

Вариант 2: основание 5

\( 1/5 = 5^{-1} \)

\( 25 = 5^2 \)

Неравенство принимает вид:

\[ (5^{-1})^{4x-6} < 5^2 \]\[ 5^{-4x+6} < 5^2 \]

Так как основание степени \( 5 > 1 \), знак неравенства сохраняется:

\[ -4x + 6 < 2 \]\[ -4x < 2 - 6 \]\[ -4x < -4 \]

Делим на \( -4 \) и меняем знак неравенства:

\[ x > 1 \] Oops, I made a mistake in the previous calculation. Let me recheck. When dividing by a negative number, the inequality sign must be reversed. So, -4x < -4 should result in x > 1. However, if we look at the first method, we got x > 2. Let me re-examine the problem again. Ah, I see. It's 1/5. So the base is less than 1. When the base is less than 1, the inequality sign flips. So, 4x - 6 > 2 is correct. Therefore, 4x > 8, and x > 2. The second method also needs to be checked again. 5^-4x+6 < 5². Since the base is 5 > 1, the inequality sign does not flip. So, -4x + 6 < 2, which means -4x < -4, and indeed x > 1. There is a contradiction. Let me check the original problem again. (1/5)^4x-6 < 25. It seems I have made a mistake. Let's go back to the first method which seems more straightforward. Base is 1/5. So 4x-6 should be GREATER than 2. So x > 2. Let me re-evaluate the second method. 5^-4x+6 < 5². Yes, base is 5. So -4x+6 < 2. -4x < -4. x > 1. This is a clear contradiction. Let me review the properties of exponents and inequalities. If a > 1, then a^m < aⁿ implies m < n. If 0 < a < 1, then a^m < aⁿ implies m > n. My first method used base 1/5, which is between 0 and 1. So, (1/5)^4x-6 < (1/5)² implies 4x - 6 > 2. This gives 4x > 8, so x > 2. My second method used base 5, which is greater than 1. So, 5^-4x+6 < 5² implies -4x + 6 < 2. This gives -4x < -4, so x > 1. I must have made a mistake in converting 25 to a power of 5 or 1/5. 25 is indeed 5². And 1/5 is 5^-1. So (1/5)^4x-6 is (5^-1)^4x-6 = 5^-(4x-6) = 5^-4x+6. So the inequality is 5^-4x+6 < 5². Since the base 5 > 1, we have -4x + 6 < 2. This leads to -4x < -4, and dividing by -4 (and reversing the sign), we get x > 1. Now let me recheck the first method. (1/5)^4x-6 < 25. 25 = 1/(1/25) = (1/5)^-2. So, (1/5)^4x-6 < (1/5)^-2. Since the base 1/5 is less than 1, we must reverse the inequality sign: 4x - 6 > -2. This gives 4x > -2 + 6, so 4x > 4, which means x > 1. Okay, both methods now consistently give x > 1. I apologize for the confusion. The previous calculation for the first method was incorrect. It should be \(25 = \frac{1}{(1/5)^2} = (1/5)^{-2}\). So, \((1/5)^{4x-6} < (1/5)^{-2}\). Since \(0 < 1/5 < 1\), we have \(4x-6 > -2\). \(4x > 4\). \(x > 1\).

Ответ: x > 1.