Question

Решить логарифмическое неравенство методом интервалов:

Answer 1

Решение на фото, надеюсь видно.

Answer 2

Решение во вложениях. Вроде ничего не напутала.

Answer 3

Solving a Logarithmic Inequality Using Interval Method

To solve a logarithmic inequality using the interval method, we need to follow a few steps. Let's solve the given logarithmic inequality step by step.

The given inequality is:

\(log_3x = t\)

\(t^2 - 10t \geq -21\)

To solve this inequality, we can rewrite it as a quadratic equation:

\(t^2 - 10t + 21 \geq 0\)

Now, let's solve this quadratic equation to find the values of \(t\) that satisfy the inequality.

To solve the quadratic equation \(t^2 - 10t + 21 = 0\), we can use the quadratic formula:

\(t = \frac{-b \pm \sqrt{D}}{2a}\)

where \(a = 1\), \(b = -10\), and \(c = 21\).

Let's calculate the discriminant \(D\) first:

\(D = b^2 - 4ac = (-10)^2 - 4(1)(21) = 100 - 84 = 16\)

Since the discriminant \(D\) is positive, we have two distinct real solutions for \(t\):

\(t_1 = \frac{-b + \sqrt{D}}{2a} = \frac{10 + 4}{2} = 7\)

\(t_2 = \frac{-b - \sqrt{D}}{2a} = \frac{10 - 4}{2} = 3\)

Now, let's analyze the intervals to determine the values of \(t\) that satisfy the inequality.

1. For \(t < 3\): - In this interval, \(t\) is less than 3. Plugging in a value less than 3 into the inequality, we get: \(t^2 - 10t + 21 \geq 0\) - Substituting \(t = 2\) into the inequality, we get: \(2^2 - 10(2) + 21 = 4 - 20 + 21 = 5 > 0\) - Therefore, the inequality is satisfied for \(t < 3\).

2. For \(3 \leq t \leq 7\): - In this interval, \(t\) is between 3 and 7 (inclusive). Plugging in a value between 3 and 7 into the inequality, we get: \(t^2 - 10t + 21 \geq 0\) - Substituting \(t = 5\) into the inequality, we get: \(5^2 - 10(5) + 21 = 25 - 50 + 21 = -4 < 0\) - Therefore, the inequality is not satisfied for \(3 \leq t \leq 7\).

3. For \(t > 7\): - In this interval, \(t\) is greater than 7. Plugging in a value greater than 7 into the inequality, we get: \(t^2 - 10t + 21 \geq 0\) - Substituting \(t = 8\) into the inequality, we get: \(8^2 - 10(8) + 21 = 64 - 80 + 21 = 5 > 0\) - Therefore, the inequality is satisfied for \(t > 7\).

Based on the analysis of the intervals, we can conclude that the values of \(t\) that satisfy the inequality \(t^2 - 10t \geq -21\) are:

- For \(t < 3\), the inequality is satisfied. - For \(3 \leq t \leq 7\), the inequality is not satisfied. - For \(t > 7\), the inequality is satisfied.

Please note that the solution above is based on the given inequality and the analysis of the intervals.