Question

Решить неравенство Х^2 -7х -30>=0

Answer 1

ответ..................

Answer 2

To solve the inequality $x^2 - 7x - 30 \geq 0$, we can use a method called the "sign chart" or "interval notation." Here's how it's done:

Step 1: Find the critical points by solving the equation $x^2 - 7x - 30 = 0$ to find the roots or zeros of the quadratic function.

Step 2: Plot the critical points on the number line.

Step 3: Test each interval defined by the critical points by picking a value within each interval and checking whether the expression $x^2 - 7x - 30$ is positive or negative.

Step 4: Determine the solution based on the signs within each interval.

Let's go through these steps:

Step 1: Find the critical points. The quadratic equation $x^2 - 7x - 30 = 0$ can be factored as $(x - 10)(x + 3) = 0$. Setting each factor to zero and solving for $x$: $x - 10 = 0 \implies x = 10$ $x + 3 = 0 \implies x = -3$

Step 2: Plot the critical points on the number line. We have two critical points: -3 and 10.

Number line: (-∞)---(-3)---(10)---(∞)

Step 3: Test intervals. Now, we will pick a value within each interval and check the sign of the expression $x^2 - 7x - 30$.

For $x < -3$, let's pick $x = -4$. $x^2 - 7x - 30 = (-4)^2 - 7(-4) - 30 = 16 + 28 - 30 = 14$, which is positive.

For $-3 < x < 10$, let's pick $x = 0$. $x^2 - 7x - 30 = 0^2 - 7(0) - 30 = -30$, which is negative.

For $x > 10$, let's pick $x = 11$. $x^2 - 7x - 30 = 11^2 - 7(11) - 30 = 121 - 77 - 30 = 14$, which is positive.

Step 4: Determine the solution. Since the inequality is $x^2 - 7x - 30 \geq 0$, we are looking for the intervals where the expression is non-negative (positive or zero).

The solution is $x \in (-∞, -3] \cup [10, ∞)$. In interval notation, the solution is $(-∞, -3] \cup [10, ∞)$.