Homework 1

1. Let $p$ and $q$ be integers and suppose that $p + 4q = 1$ and $q - 1 = 2$. Show that $p = -11$.

2. Let $a$ and $b$ be real numbers and suppose that $a + 2b = 4$ and $a - b = 1$. Show that $a = 2$.

3. Let $x$ be an integer, with $x \geq 9$. Show that $x^3 - 8x^2 + 2x \geq 3$.

4. Let $x$ be a rational number. Show that $x^2 - 2x \ge -1$.

5. Let $a$ and $b$ be real numbers and suppose that $-b \le a \le b$. Show that $a^2 \le b^2$.
   This problem is solved in words in the text in Example 2.1.7, so don't submit a solution in words, only in Lean.