Question

Log(1/2) |x| >= |x| - 1

Answer 1

$\log_{0,5}|x| \geq |x| - 1$

ОДЗ: $|x| \neq 0; \ x \neq 0$

Решим данное неравенство графически.

Рассмотрим две функции $y_{1} = \log_{0,5}|x|$ и $y_{2} = |x| - 1$ и изобразим их на координатной плоскости (см. вложение).

Из графика видно, что логарифмическая функция не меньше модульной на промежутке $x \in [-1; \ 0) \cup (0; \ 1]$

Ответ: $x \in [-1; \ 0) \cup (0; \ 1]$

Answer 2

To solve the inequality Log(1/2) |x| >= |x| - 1, we need to consider different cases based on the value of x. Remember that Log(1/2) is a negative value, approximately equal to -0.3010.

Case 1: x > 0

In this case, |x| is equal to x, so the inequality becomes:

Log(1/2) x >= x - 1

To simplify the inequality, let's move x to one side:

Log(1/2) x - x >= -1

Now, factor out x:

x (Log(1/2) - 1) >= -1

Since Log(1/2) is approximately -0.3010, we have:

x (-0.3010 - 1) >= -1

x (-1.3010) >= -1

Now, divide both sides by -1.3010 (note that we are dividing by a negative number, so the inequality sign flips):

x <= -1 / 1.3010

x <= -0.768

So, for x > 0, the solution is x <= -0.768.

Case 2: x < 0

In this case, |x| is equal to -x, so the inequality becomes:

Log(1/2) (-x) >= -x - 1

To simplify the inequality, let's move -x to one side:

Log(1/2) (-x) + x >= -1

Now, factor out x:

x (1 - Log(1/2)) >= -1

Since Log(1/2) is approximately -0.3010, we have:

x (1 + 0.3010) >= -1

x (1.3010) >= -1

Now, divide both sides by 1.3010:

x >= -1 / 1.3010

x >= -0.768

So, for x < 0, the solution is x >= -0.768.

Combining the results from both cases, we get the final solution:

x <= -0.768 or x >= -0.768