Question
6. a) \( \frac{\log _{0,5}(2-3 x)}{\log _{0,5} 4}>0 \);
Ask by Owen Bryant. in Turkmenistan
Dec 03,2024
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Answer
Решение неравенства: \( \frac{1}{3} < x < \frac{2}{3} \).
Solution
Для решения неравенства \( \frac{\log_{0.5}(2-3x)}{\log_{0.5} 4} > 0 \) начнем с анализа логарифмов.
1. **Определим знак логарифма**:
- Логарифм с основанием \( 0.5 \) (меньше 1) меняет знак. То есть:
- \( \log_{0.5}(2-3x) > 0 \) тогда и только тогда, когда \( 2-3x < 1 \).
- \( \log_{0.5}(4) > 0 \) (так как \( 4 > 1 \)).
2. **Решим неравенство**:
- Для \( \log_{0.5}(2-3x) > 0 \):
\[
2 - 3x < 1 \implies -3x < -1 \implies x > \frac{1}{3}.
\]
3. **Условия для логарифма**:
- Также необходимо, чтобы \( 2 - 3x > 0 \):
\[
2 - 3x > 0 \implies 2 > 3x \implies x < \frac{2}{3}.
\]
4. **Объединим условия**:
- Мы имеем два условия:
- \( x > \frac{1}{3} \)
- \( x < \frac{2}{3} \)
Таким образом, решение неравенства:
\[
\frac{1}{3} < x < \frac{2}{3}.
\]
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Simplify this solution Bonus Knowledge
To evaluate the inequality, we can start by simplifying it. Since \(\log_{0.5}(4) = -2\) (because \((0.5)^{-2}=4\)), we can rewrite the inequality as: \(\log_{0.5}(2-3x) > 0\). This implies that \(2-3x < 1\) because a logarithm to a base less than 1 is negative when its argument is greater than 1. Simplifying gives \(3x > 1\) or \(x > \frac{1}{3}\). Hence, the solution set for the inequality is \(x < \frac{2}{3}\) and \(x > \frac{1}{3}\). A combination of these gives the final solution as \((\frac{1}{3}, \frac{2}{3})\). This means any \(x\) within this range will satisfy the original inequality!
