Let's work through the problems from this image step-by-step.

Problem 7.15 Analysis:

Given Information: A real number $x$ satisfies $|x - 4| < 1$.

This inequality implies: $3 < x < 5$

Part (a):

To prove: $\frac{1}{3} < \frac{x^2}{x + 22} < 1$

Since $x$ is between 3 and 5, we need to evaluate $\frac{x^2}{x + 22}$ for values within this range and show that it remains between $\frac{1}{3}$ and $1$.

Part (b):

Define: $a = | |x - 3| - 2 | + | |x - 5| - 2 |$

1. Part (i): Prove that the value of $a$ is independent of $x$.

   To address this, substitute values of $x$ within the interval $3 < x < 5$ and simplify the expression to see if $a$ has a constant value within this range.

2. Part (ii): Given conditions

   • $|\beta - \gamma| = |\gamma - \delta| - a$
   • $|\beta - \varepsilon| = a^2 - |\varepsilon - \delta|$

   We are to prove: $2 \leq |\beta - \delta| \leq 4$

   This part requires careful manipulation of absolute values and verification based on the given inequalities and conditions.

Would you like me to go into detailed steps for each of these parts, or is there a specific section where you need clarification?

Additional Questions for Exploration:

1. How would the range for $\frac{x^2}{x + 22}$ change if $x$ were in a different interval?
2. Can you find another way to simplify the expression for $a$ to see if it has a constant value?
3. What implications does the interval $|x - 4| < 1$ have on $x^2$ or other expressions involving $x$?
4. How can we verify the conditions in part (ii) algebraically with specific values?
5. Would changing the constants in the inequality (e.g., $1/3$ and $1$) affect the solution approach in part (a)?

Tip:

When working with absolute value inequalities, consider simplifying by analyzing specific intervals based on the expressions inside the absolute values. This can often make the calculations easier.