1. **Find the interval of $x$ satisfying:** $$| |x - 3| - 2 | \geq 1$$ **Step 1:** Let $y = |x - 3|$. Then the inequality becomes: $$|y - 2| \geq 1$$ **Step 2:** The absolute value inequality $|A| \geq B$ means $A \leq -B$ or $A \geq B$. So: $$y - 2 \leq -1 \quad \text{or} \quad y - 2 \geq 1$$ **Step 3:** Solve each: - $y - 2 \leq -1 \Rightarrow y \leq 1$ - $y - 2 \geq 1 \Rightarrow y \geq 3$ **Step 4:** Recall $y = |x - 3|$, so: $$|x - 3| \leq 1 \quad \text{or} \quad |x - 3| \geq 3$$ **Step 5:** Solve each absolute value inequality: - $|x - 3| \leq 1 \Rightarrow 2 \leq x \leq 4$ - $|x - 3| \geq 3 \Rightarrow x \leq 0 \quad \text{or} \quad x \geq 6$ **Answer:** $$(-\infty, 0] \cup [2, 4] \cup [6, \infty)$$ --- 2. **Prove for all real $x,y$:** $$|||x| - |y|| \leq |x - y|$$ **Step 1:** This is the reverse triangle inequality. **Step 2:** By the triangle inequality: $$|x| = |(x - y) + y| \leq |x - y| + |y|$$ Rearranged: $$|x| - |y| \leq |x - y|$$ **Step 3:** Similarly, $$|y| = |(y - x) + x| \leq |y - x| + |x| = |x - y| + |x|$$ Rearranged: $$|y| - |x| \leq |x - y|$$ **Step 4:** Combining both: $$-|x - y| \leq |x| - |y| \leq |x - y|$$ Taking absolute value: $$|||x| - |y|| \leq |x - y|$$ --- 3. **Prove by induction:** $$2 + 4 + 6 + \cdots + 2n = n(n + 1), \quad n \geq 1$$ **Step 1:** Base case $n=1$: $$2 = 1 \times (1 + 1) = 2$$ True. **Step 2:** Assume true for $n=k$: $$2 + 4 + \cdots + 2k = k(k + 1)$$ **Step 3:** For $n = k+1$: $$2 + 4 + \cdots + 2k + 2(k+1) = k(k + 1) + 2(k + 1)$$ $$= (k + 1)(k + 2)$$ **Step 4:** Thus, true for $k+1$. By induction, the formula holds for all $n \geq 1$. --- 4. **Use Extended Euclidean Algorithm to find integers $x,y$ such that:** $$\gcd(128, 536) = 128x + 536y$$ **Step 1:** Compute gcd using Euclidean algorithm: $$536 = 128 \times 4 + 24$$ $$128 = 24 \times 5 + 8$$ $$24 = 8 \times 3 + 0$$ So, $\gcd(128, 536) = 8$. **Step 2:** Back substitute to express 8 as linear combination: $$8 = 128 - 24 \times 5$$ But $24 = 536 - 128 \times 4$, so: $$8 = 128 - (536 - 128 \times 4) \times 5 = 128 - 536 \times 5 + 128 \times 20 = 128 \times 21 - 536 \times 5$$ **Answer:** $$x = 21, \quad y = -5$$ --- **Summary:** 1. Interval: $(-\infty, 0] \cup [2,4] \cup [6, \infty)$ 2. Inequality holds by reverse triangle inequality. 3. Induction proves sum formula. 4. $\gcd(128,536) = 8 = 128 \times 21 - 536 \times 5$.