Question

Упростить неравенство:(a+x/x-2x/x-a)^2*x-a/a+x^2

Answer 1

Дальше не знаю как :(

Answer 2

I can understand and communicate in Russian. I will try to help you with your query.

To simplify the inequality, we need to do the following steps:

- Simplify the fraction inside the parentheses by finding the common denominator and subtracting the numerators. - Square the simplified fraction and multiply it by the term x - a. - Simplify the fraction in the denominator by finding the common denominator and adding the numerators. - Divide the numerator and the denominator by the common factor x - a. - Solve the resulting quadratic inequality by finding the roots and the sign of the leading coefficient.

The detailed solution is:

(a + x)/(x - 2x/(x - a))^2 * (x - a)/(a + x^2) < 0

(a + x)/((x - a - 2x)/(x - a))^2 * (x - a)/(a + x^2) < 0

(a + x)/((x - 3x + a)/(x - a))^2 * (x - a)/(a + x^2) < 0

(a + x)/((a - 2x)/(x - a))^2 * (x - a)/(a + x^2) < 0

(a + x) * (x - a)^2/((a - 2x)^2 * (x - a)) * (x - a)/(a + x^2) < 0

(a + x) * (x - a)/((a - 2x)^2 * (a + x^2)) < 0

(x - a)/((a - 2x)^2 * (a + x^2)) < 0

The roots of the inequality are a, 0, 2a, and -a. The leading coefficient is positive, so the inequality is satisfied when the expression is negative. This happens when x is between 0 and a, or between -a and 2a. Therefore, the solution set is:

(-a, 0) U (0, a) U (a, 2a)