Question

Оформить решение неравенства используя символику теории множеств (3x+2) (x-4)<0

Answer 1

$(3x+2) (x-4)\ \textless \ 0$

$3x+2=0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~x-4=0 \\ 3x=-2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~x=4\\ x=- \frac{2}{3}$

Наносим корни на интервалы (смотри рисунок) и выставляем знаки.

Ответ: $x$ ∈ $(- \frac{2}{3} ;~4)$

Answer 2

Solving the Inequality Using Set Theory Symbolism

To solve the inequality (3x + 2)(x - 4) < 0 using set theory symbolism, we can follow these steps:

1. Find the Critical Points: - The critical points are the values of x that make the expression (3x + 2)(x - 4) equal to zero. These points divide the number line into intervals where the inequality may change its sign.

2. Test Intervals: - We will test the intervals between the critical points to determine where the inequality is satisfied.

3. Expressing the Solution: - Finally, we will express the solution using set theory symbolism.

Finding the Critical Points

The critical points are the values of x that make the expression (3x + 2)(x - 4) equal to zero. We can find these points by setting the expression equal to zero and solving for x.

The critical points are: - x = -2/3 (from the factor 3x + 2 = 0) - x = 4 (from the factor x - 4 = 0)

Testing Intervals

We will test the intervals created by the critical points to determine where the inequality is satisfied. We can use test points within each interval to determine the sign of the expression (3x + 2)(x - 4).

- For the interval x < -2/3, we can choose x = -1 as a test point. - For the interval -2/3 < x < 4, we can choose x = 0 as a test point. - For the interval x > 4, we can choose x = 5 as a test point.

Expressing the Solution Using Set Theory Symbolism

Based on the tests, we can express the solution using set theory symbolism. The solution to the inequality (3x + 2)(x - 4) < 0 in set theory symbolism is:

{-2/3 < x < 4}

This represents the values of x for which the inequality is satisfied.

I hope this helps! If you have further questions or need additional assistance, feel free to ask.