Algebra

We are given the inequality: and the condition .

1. Step 1: Substituting the values for : - For the lower bound, substitute into the inequality: - For the upper bound, substitute into the inequality: Thus, we get:

2. Step 2: Finding : From the inequality, we can see that the upper bound must be for the condition to hold true for all in the interval . Therefore, the value of is .

We are given the equation: Step 1: Multiply the entire equation by the LCM of 9 and 8, which is 72.
Multiplying both sides by 72 to eliminate the denominators: Simplifying:
Step 2: Expand both squared terms.
First, expand and : Now, substitute these expansions into the equation:
Step 3: Distribute the constants 8 and 9.
Distribute 8 and 9 to get:
Step 4: Combine like terms.
Now, combine like terms:
Step 5: Simplify the equation.
Now, subtract 72 from both sides:
Step 6: Solve the quadratic equation.
Now, solve the quadratic equation using the quadratic formula: For the equation , we have , , and . Substituting these values into the quadratic formula: Since the discriminant is negative, there are no real solutions. % Final Answer
Final Answer: \text{No real solutions.}

Concept:
Complex Numbers - Properties of Conjugates.
For any complex number :

• If , then is purely real.
• If , then is purely imaginary.

Step 1: Analyze the expression for .
Let . The conjugate of is .
We can rewrite in terms of : $ z_1 \bar{z}_1 = z_1 z_1 z_2 u = 3+2i u \bar{u} = 3-2i z_2 u z_2 \bar{z}_2 = -z_2 z_2$ is a purely imaginary number.

Concept:
Complex Numbers - Modulus Properties.
The modulus of a quotient is the quotient of the moduli: .
The modulus of a complex number is .
Step 1: Evaluate the modulus of the first fraction.
Let's find the modulus of the numerator and denominator separately. $ 1 1 $

Concept:
Complex Numbers - Multiplication of a Complex Number and a Conjugate.
Step 1: Identify the values and their conjugates.
Given , its conjugate is .
Given (rewriting in standard form), its conjugate is .
Step 2: Evaluate the term .
Multiply and : $ i^2 = -1 z_2\overline{z_1} z_2 \overline{z_1} i^2 = -1 (z_1\overline{z_2} + z_2\overline{z_1}) + (z_1\overline{z_2} + z_2\overline{z_1}) $

Concept:
Complex Numbers - Powers of and Arithmetic Progressions.
Step 1: Combine the terms using exponent rules.
When multiplying terms with the same base, you add their exponents: $ n n = 2026 S_n = \frac{n(n+1)}{2} i i^1 = i i^2 = -1 i^3 = -i i^4 = 1 (1013 \times 2027) \pmod 4 4k + 3 i^3 = i^2 \times i = (-1) \times i = -i -i$.

Concept:
Functions - Domain and Range of Inverse Functions.
Step 1: Recall the relationship between a function and its inverse.
The domain of an inverse function is exactly equal to the range of the original function .
Therefore, our objective is to find the range of for the given domain .
Step 2: Rewrite the function by completing the square.
Given .
Rewrite it as .
$ (-1, 2) x x \le -1 (x + 1)^2 \ge 0 x = -1 x -1 -2, -3, -\infty (x+1)^2 2 f(x) [2, \infty) (f^{-1}) = (f) [2, \infty)$.

Concept:
Functions - Composition of Functions and Involutions.
Step 1: Identify the function form and objective.
We are given .
We need to find the value of , which means applying the function 1000 times consecutively.
Step 2: Evaluate the first composition .
Substitute into itself: $ x f(f(x)) f^2(x) = x f^{1000}(100) f^2(x) = x x f^{1000}(x) = x f^{1000}(100) = 100$.