Question

11ˣ - 6 - (24*11ˣ - 244)/(121ˣ - 16*11ˣ + 60) ≤ 1/(11ˣ - 10)

Answer 1

11^x=a
(a-6)-(24a-244)/(a²-16a+60)-1/(a-10)≤0
(a-6)-(24a-244)/(a-6)(a-10)-1/(a-10)≤0
(a²-12a+36-24a+244-a+6)/(a-6)(a-10)≤0
(a²-37a+286)/(a-6)(a-10)≤0
a1+a2=37 U a1*a2=286
a1=11 U a2=26
a=6
a=10
           +              _                  +                _                +
-------------(6)------------(10)-----------[11]------------[26]-----------------
6<a<10⇒6<11^x<10⇒log(11)6<x<log(11)6
11≤a≤26⇒11≤11^x≤26⇒1≤x≤log(11)26
Ответ x∈(log(11)6;log(11)10) U [1;log(11)26]

Answer 2

To solve the inequality 11^x - 6 - (2411^x - 244)/(121^x - 1611^x + 60) ≤ 1/(11^x - 10), you can start by simplifying and rearranging the terms to isolate the variable x. Here's a step-by-step solution:

1. First, let's deal with the denominator on the right side: 1/(11^x - 10).

2. Next, multiply both sides of the inequality by (11^x - 10) to eliminate the denominator on the right side:

   (11^x - 6 - (2411^x - 244)/(121^x - 1611^x + 60))(11^x - 10) ≤ 1

3. Now, expand the left side of the inequality:

   (11^x - 6 - (2411^x - 244)/(121^x - 1611^x + 60))(11^x) - (11^x - 6 - (2411^x - 244)/(121^x - 1611^x + 60))(10) ≤ 1

4. Distribute the terms on both sides of the inequality:

   11^(2x) - 611^x - 1011^x + 2411^x - 244 - (2411^(2x) - 24011^x - 1011^x + 600) ≤ 1

5. Simplify further:

   11^(2x) - 1611^x - 244 - 2411^(2x) + 25011^x + 1011^x - 600 ≤ 1

6. Combine like terms:

   -1311^(2x) + 24411^x - 844 ≤ 1

7. Move 1 to the left side of the inequality:

   -1311^(2x) + 24411^x - 845 ≤ 0

Now you have a quadratic inequality in terms of 11^x. To solve it, you can use a quadratic formula or graphing methods. However, I can't provide a specific numerical solution without more information about the value of x you're interested in. If you have a specific value of x in mind or want to solve it for a range of x values, you can use a calculator or a computer algebra system to find the solutions.