Question

Решить неравенство: log2(4x^2-1)-log2(x)≤log2(5x+9/x-11)

Answer 1

$log_{2}(4x^2-1)-log_{2}(x) \leq log_{2}(5x+ \frac{9}{x} -11) \\ \\O DZ: \\ \\ 4x^2-1\ \textgreater \ 0 \\ x\ \textgreater \ 0 \\ 5x+ \frac{9}{x} -11\ \textgreater \ 0 \\ x \neq 0 \\ \\ (2x-1)(2x+1)\ \textgreater \ 0 \\ x\ \textgreater \ 0 \\ 5x^2-11x+9\ \textgreater \ 0;D=121-4*45=121-180=-35\ \textless \ 0; \\ \\ x\in(-oo;-1/2)U(1/2;+oo) \\ x\n(0;+oo) \\ \\ ODZ:x\in(1/2;+oo)$
2>1, знак сохраняем

$\frac{4x^2-1}{x} \leq 5x+ \frac{9}{x} -11 \\ \\ \frac{4x^2-1}{x} \leq \frac{5x^2-11x+9}{x} \\ \\ \frac{5x^2-11x+9-4x^2+1}{x} \geq 0 \\ \\ \frac{x^2-11x+10}{x} \geq 0 \\ \\ D=121-40=81=9^2; x_{1} =10; x_{2} =1 \\ \\ \frac{(x-10)(x-1)}{x} \geq 0 \\ \\ ---(0)+++[1]---[10]++++ \\ \\ x\in(0;1]U[10;+oo)$

с учетом ОДЗ,получаем ответ:
$x\in( \frac{1}{2} ;1]U[10;+oo)$

Answer 2

To solve the inequality log2(4x^2 - 1) - log2(x) ≤ log2(5x + 9/x - 11), we need to follow certain steps. Remember that for logarithms to be defined, the arguments must be positive.

Step 1: Combine the logarithms using logarithm rules. Step 2: Simplify the expression to a single logarithm if possible. Step 3: Eliminate the logarithm and solve the resulting inequality. Step 4: Check the validity of the solutions.

Let's proceed with the solution:

Step 1: Combine the logarithms using logarithm rules. log2(4x^2 - 1) - log2(x) ≤ log2(5x + 9/x - 11)

Step 2: Simplify the expression to a single logarithm if possible. Apply the quotient rule for logarithms, which states that log_a(b) - log_a(c) = log_a(b/c): log2((4x^2 - 1)/x) ≤ log2(5x + 9/x - 11)

Step 3: Eliminate the logarithm and solve the resulting inequality. To eliminate the logarithm, we can take both sides as the base 2 exponent:

(4x^2 - 1)/x ≤ (5x + 9/x - 11)

Step 4: Solve the inequality:

First, let's get rid of the fraction by multiplying both sides by x:

x(4x^2 - 1) ≤ x(5x + 9/x - 11)

Simplify:

4x^3 - x ≤ 5x^2 + 9 - 11x

Now, bring all terms to one side:

4x^3 - 5x^2 - x + 11 ≤ 9

Finally, set the inequality to zero:

4x^3 - 5x^2 - x + 2 ≤ 0

We need to solve this cubic inequality for x. The solution to cubic inequalities can be complex and involve different intervals. For a quick overview, we can observe that the polynomial has roots at x = 1/2, x = -1, and x = 2. These roots divide the real number line into four intervals: (-∞, -1), (-1, 1/2), (1/2, 2), and (2, ∞).

Now, we can test each interval to find the valid solution set:

1. For x < -1: Pick x = -2 (a value less than -1): 4(-2)^3 - 5(-2)^2 - (-2) + 2 = -30 (negative)

2. For -1 < x < 1/2: Pick x = 0 (a value between -1 and 1/2): 4(0)^3 - 5(0)^2 - (0) + 2 = 2 (positive)

3. For 1/2 < x < 2: Pick x = 1 (a value between 1/2 and 2): 4(1)^3 - 5(1)^2 - (1) + 2 = 0 (non-negative)

4. For x > 2: Pick x = 3 (a value greater than 2): 4(3)^3 - 5(3)^2 - (3) + 2 = 34 (positive)

From the above tests, we can see that the inequality is satisfied for the interval -1 < x < 1/2 and the interval 1/2 < x < 2.

Thus, the solution to the original inequality is: -1 < x < 1/2 OR 1/2 < x < 2.