Homework 2

1. Let $x$ be a rational number whose square is 9, and which is greater than 1. Show that $x=3$.

   Suggested steps: Prove that $x(x+3)=3(x+3)$, then cancel to deduce that $x=3$.

2. Let $s$ be a rational number for which $3s \leq -15$ and $2s \geq -10$. Show that $s=-5$.

   You will probably use the lemma le_antisymm, stating if $x\leq y$ and $x\geq y$ then $x = y$.

3. Let $t$ be a rational number and suppose that $t=2$ or $t=-3$. Show that $t^2+t-6=0$.

4. Let $x$ be any integer. Show that $3x \neq 10$.

   You will probably use the lemma le_or_succ_le, stating that for integers $x$, $y$ either $x\leq y$ or $x \geq y+1$.

5. Let $x$ and $y$ be real numbers, at least one of which is greater than or equal to 2. Suppose also that $x^2+y^2=4$. Show that $x^2y^2=0$.

   You will probably use the lemma le_antisymm, stating if $x\leq y$ and $x\geq y$ then $x = y$.