Question

0,3^x^2-14x ⩾ (3 1/3)^24

Answer 1

Ответ:

х€[2;12]

Объяснение:

${0.3}^{ {x}^{2} - 14x} \geqslant {(3 \frac{1}{3}) }^{24}$

${0.3}^{ {x}^{2} - 14x} = {( \frac{3}{10})}^{ {x}^{2} - 14x}$

${(3 \frac{1}{3})}^{24}={( \frac{10}{3}) }^{24} ={ {(( \frac{3}{10})}^{ - 1}})^{24} =$

${( \frac{3}{10})}^{ {x}^{2} - 14x} \geqslant {( \frac{3}{10})}^{ - 24}$

основание степени а =3/10,

$0 < \frac{3}{10} < 1$

знак неравенства меняем:

${x}^{2} - 14x \leqslant - 24$

${x}^{2} - 14x + 24 \leqslant 0$

метод интервалов:

х1=2, х2=12

(х-2)×(х-12)<=0

+++++ [2] --------- [12] ++++++> х

х€[2;12]

Answer 2

To solve the inequality 0.3^(x^2 - 14x) ≥ (3^(1/3))^24, you can follow these steps:

1. Simplify both sides of the inequality: 0.3^(x^2 - 14x) ≥ (3^(1/3))^24

   0.3^(x^2 - 14x) ≥ 3^8

2. Take the natural logarithm (ln) of both sides of the inequality: ln(0.3^(x^2 - 14x)) ≥ ln(3^8)

3. Use the properties of logarithms to simplify the left side: (x^2 - 14x) * ln(0.3) ≥ 8 * ln(3)

4. Now, divide both sides by ln(0.3) to isolate the x terms: x^2 - 14x ≥ (8 * ln(3)) / ln(0.3)

5. At this point, you can solve the quadratic inequality. However, it's important to note that the inequality may have multiple solutions. Let's first find the critical points by setting the left side equal to zero:

   x^2 - 14x = 0

   Factor the left side: x(x - 14) = 0

   This gives two critical points: x = 0 and x = 14.

6. Now, analyze the inequality in intervals separated by these critical points.

   Interval 1: x < 0 In this interval, x^2 - 14x is positive, and ln(0.3) is negative. Therefore, the inequality becomes:

   Positive * Negative ≥ Positive This inequality is satisfied for all x < 0.

   Interval 2: 0 < x < 14 In this interval, x^2 - 14x is positive, and ln(0.3) is negative. Therefore, the inequality becomes:

   Positive * Negative ≥ Positive This inequality is satisfied for all 0 < x < 14.

   Interval 3: x > 14 In this interval, x^2 - 14x is positive, and ln(0.3) is negative. Therefore, the inequality becomes:

   Positive * Negative ≥ Positive This inequality is satisfied for all x > 14.

So, the solution to the inequality is the union of all three intervals:

x < 0 or 0 < x < 14 or x > 14

These are the values of x that satisfy the given inequality.