Question

Каким методом решать неравенство х-4х^2/x-1>0 ? Как называется неравенство такого вида? Заранее большое спасибо!

Answer 1

Методом интервала надо решать

Answer 2

Solving the Inequality and Its Name

To solve the inequality x - 4x^2 / (x - 1) > 0, we can follow these steps:

1. Find the Critical Points: The critical points are the values of x for which the expression x - 4x^2 / (x - 1) is undefined or equal to 0. These critical points will divide the number line into intervals.

2. Test Intervals: Test the intervals defined by the critical points to determine where the inequality is satisfied.

3. Identify the Name of the Inequality: This type of inequality is known as a rational inequality due to the presence of a rational expression.

Now, let's proceed with solving the inequality and identifying the type of inequality.

Solving the Inequality

First, let's find the critical points by setting the denominator and the numerator equal to 0:

1. Find the Critical Points: - Set the denominator (x - 1) equal to 0: x - 1 = 0. Solving for x gives us x = 1. - Set the numerator (x - 4x^2) equal to 0: x - 4x^2 = 0. Factoring out an x gives us x(1 - 4x) = 0. So, the critical points are x = 0 and x = 1/4.

2. Test Intervals: - We have three intervals to test: (-∞, 0), (0, 1/4), and (1/4, ∞). - Test a value from each interval in the original inequality x - 4x^2 / (x - 1) to determine where it is greater than 0.

- For x = -1, x - 4x^2 / (x - 1) = -1 - 4(-1)^2 / (-1 - 1) = -1 + 4 / -2 = -1 + (-2) < 0. - For x = 1/8, x - 4x^2 / (x - 1) = 1/8 - 4(1/8)^2 / (1/8 - 1) = 1/8 - 1/2 / (-7/8) = 1/8 - 1/2 * (-8/7) = 1/8 + 4/7 > 0. - For x = 2, x - 4x^2 / (x - 1) = 2 - 4*2^2 / (2 - 1) = 2 - 16 / 1 = 2 - 16 < 0.

3. Identify the Name of the Inequality: - The type of inequality we have solved is a rational inequality due to the presence of a rational expression.

Therefore, the solution to the inequality x - 4x^2 / (x - 1) > 0 is x ∈ (0, 1/4) ∪ (1, ∞).