Question

Решите систему неравенств {х²-4х≥0 {9-х²>0​ Смотреть ответ kazemir4ik kazemir4ik Объяснение: Можно еще нарисовать общую прямую с точками из обоих неравенств

Answer 1

Solving the System of Inequalities

To solve the system of inequalities x² - 4x ≥ 0 and 9 - x² > 0, we need to find the values of x that satisfy both inequalities.

Let's solve each inequality separately and then find the intersection of their solution sets.

Solving the Inequality x² - 4x ≥ 0

To solve the inequality x² - 4x ≥ 0, we can factor it as follows:

x(x - 4) ≥ 0

To find the values of x that satisfy this inequality, we need to consider three cases:

1. When x > 0, both x and (x - 4) are positive, so the inequality is satisfied. 2. When x < 0, both x and (x - 4) are negative, so the inequality is satisfied. 3. When x = 0, the inequality is also satisfied.

Therefore, the solution to the inequality x² - 4x ≥ 0 is x ≤ 0 or x ≥ 4.

Solving the Inequality 9 - x² > 0

To solve the inequality 9 - x² > 0, we can rearrange it as follows:

-x² + 9 > 0

To find the values of x that satisfy this inequality, we can factor it as a difference of squares:

-(x - 3)(x + 3) > 0

To determine the sign of the expression -(x - 3)(x + 3), we can use a sign chart or test values from each interval.

1. When x < -3, both (x - 3) and (x + 3) are negative, so the inequality is satisfied. 2. When -3 < x < 3, (x - 3) is negative and (x + 3) is positive, so the inequality is not satisfied. 3. When x > 3, both (x - 3) and (x + 3) are positive, so the inequality is satisfied.

Therefore, the solution to the inequality 9 - x² > 0 is x < -3 or x > 3.

Finding the Intersection of the Solution Sets

To find the values of x that satisfy both inequalities, we need to find the intersection of the solution sets.

From the first inequality, we have x ≤ 0 or x ≥ 4. From the second inequality, we have x < -3 or x > 3.

Taking the intersection of these solution sets, we find that the values of x that satisfy both inequalities are x ≤ 0 or x ≥ 4.

Graphical Representation

To visualize the solution, we can plot the graphs of the two inequalities on a coordinate plane.

The graph of x² - 4x ≥ 0 is a parabola that opens upwards and intersects the x-axis at x = 0 and x = 4. The shaded region below the curve represents the solution set.

The graph of 9 - x² > 0 is an inverted parabola that opens downwards and intersects the x-axis at x = -3 and x = 3. The shaded region between the two vertical lines represents the solution set.

By combining the shaded regions of both graphs, we can see that the solution to the system of inequalities is the region where the shaded regions overlap, which is x ≤ 0 or x ≥ 4.

I hope this helps! Let me know if you have any further questions.