Question

(1 + 5х)( х - 2) < 0

Answer 1

Ответ: на фото сверху.

Answer 2

To solve the inequality $(1 + 5x)(x - 2) < 0$, you need to find the intervals where the expression is negative. Here's how you can do it step by step:

1. Find the critical points by setting each factor equal to zero: $1 + 5x = 0$ => $x = -\frac{1}{5}$ $x - 2 = 0$ => $x = 2$

2. Use these critical points to create test intervals. These intervals are separated by the critical points. We'll test the intervals: Interval 1: $-\infty < x < -\frac{1}{5}$ Interval 2: $-\frac{1}{5} < x < 2$ Interval 3: $2 < x < \infty$

3. Choose a test point from each interval and evaluate the expression $(1 + 5x)(x - 2)$:

   • For Interval 1, choose $x = -1$: $(1 + 5(-1))(-1 - 2) = (-4)(-3) = 12$
   • For Interval 2, choose $x = 0$: $(1 + 5(0))(0 - 2) = (1)(-2) = -2$
   • For Interval 3, choose $x = 3$: $(1 + 5(3))(3 - 2) = (16)(1) = 16$

4. Analyze the results:

   • In Interval 1, the expression is positive (greater than 0).
   • In Interval 2, the expression is negative (less than 0).
   • In Interval 3, the expression is positive (greater than 0).

5. Determine the solution: The expression $(1 + 5x)(x - 2)$ is negative (less than 0) in the interval $-\frac{1}{5} < x < 2$.

So, the solution to the inequality is: $-\frac{1}{5} < x < 2$.