Question

7^x-1/8x^2<7^1-x*(7^1/8)^x^2+6

Answer 1

Имеем:
$7^x - {1/8} x^2 \ \textless \ 7^1 - x (7^{1/8})^x^2 + 6$
Преобразуем:
$7^x - x^2/8\ \textless \ 13 - 7^{x^2/8} x$
Домножим обе части на 8 и перенесем за знак неравенства:
$-x^2 + 8* 7^{x^2/8} x + 8 *7^x - 104$
А так как у нас неравенство, то решением является диапазон
от минус бесконечности до $-x^2 + 8* 7^{x^2/8} x + 8 *7^x - 104$
Численный результат  x ≈  1.24923.

Answer 2

To solve the inequality 7^(x-1)/(8x^2) < 7^(1-x) * (7^(1/8))^(x^2) + 6, we'll simplify the expression step by step. Please note that due to the complexity of the equation, it might not have an exact algebraic solution, and we may need to use numerical methods to approximate the solution.

Let's begin:

1. Rewrite the equation using exponentiation notation: 7^(x-1)/(8x^2) < 7^(1-x) * (7^(1/8))^(x^2) + 6

2. Simplify the exponents using the properties of exponents: 7^(x-1)/(8x^2) < 7^(1-x) * 7^((1/8) * x^2) + 6

3. Apply the product rule of exponents to the right side: 7^(x-1)/(8x^2) < 7^(1-x + (1/8) * x^2) + 6

4. Combine the exponents on the right side: 7^(x-1)/(8x^2) < 7^((8 - 8x + x^2)/8) + 6

5. Rewrite the inequality without using exponents: Log base 7 of [(x-1)/(8x^2)] < Log base 7 of [7^((8 - 8x + x^2)/8) + 6]

At this point, we have converted the inequality into a logarithmic equation. Now, to find the solution, we'll need to apply numerical methods such as graphing or iterative approximation since solving it algebraically is difficult.