Question

Помогите решить неравенство 5(х-3)(х-4) >0 как это изобразить на координатной прямой мы пишем 5 отмечаем или нет?

Answer 1

5 не влияет на знак неравенства, ее можно вообще не писать, и уж тем более не отмечать на прямой.
Отмечаем точки 3 и 4. Точки выколотые, т.к. неравенство строгое.
Расставляем знаки на полученных промежутках: + - +
Выбираем промежутки с "+", т.к. в неравенстве знак ">"
Ответ: x∈(-∞; 3) U (4; +∞)

Answer 2

Solving the Inequality

To solve the inequality 5(x-3)(x-4) > 0, we can use the concept of interval notation and the zero-product property. Here's how we can proceed:

1. Find the critical points by setting each factor equal to zero and solving for x: - Setting x-3 = 0, we get x = 3. - Setting x-4 = 0, we get x = 4.

2. Plot the critical points on a number line. We have x = 3 and x = 4.

3. Choose test points from each interval created by the critical points and evaluate the inequality. We can choose x = 0, x = 3.5, and x = 5 as test points.

- For x = 0: - 5(0-3)(0-4) = 5(-3)(-4) = 60, which is greater than 0. - For x = 3.5: - 5(3.5-3)(3.5-4) = 5(0.5)(-0.5) = -1.25, which is less than 0. - For x = 5: - 5(5-3)(5-4) = 5(2)(1) = 10, which is greater than 0.

4. Analyze the intervals based on the signs of the test points: - From x = 0 to x = 3, the inequality is greater than 0. - From x = 3 to x = 4, the inequality is less than 0. - From x = 4 to positive infinity, the inequality is greater than 0.

Graphing on a Coordinate Plane

To graph the inequality 5(x-3)(x-4) > 0 on a coordinate plane, we can follow these steps:

1. Draw a horizontal number line and mark the critical points x = 3 and x = 4.

2. Divide the number line into three intervals based on the critical points: (-∞, 3), (3, 4), and (4, +∞).

3. Choose a test point from each interval and evaluate the inequality.

- For the interval (-∞, 3), we can choose x = 0 as a test point. Substituting x = 0 into the inequality, we get 5(0-3)(0-4) = 60, which is greater than 0. - For the interval (3, 4), we can choose x = 3.5 as a test point. Substituting x = 3.5 into the inequality, we get 5(3.5-3)(3.5-4) = -1.25, which is less than 0. - For the interval (4, +∞), we can choose x = 5 as a test point. Substituting x = 5 into the inequality, we get 5(5-3)(5-4) = 10, which is greater than 0.

4. Based on the signs of the test points, we can determine the solution to the inequality:

- The solution is greater than 0 for the intervals (-∞, 3) and (4, +∞). - The solution is less than 0 for the interval (3, 4).

Therefore, the solution to the inequality 5(x-3)(x-4) > 0 is x < 3 or x > 4.

Please note that the critical points x = 3 and x = 4 are not included in the solution because the inequality is strict (greater than 0).