Question 101 – 1.1 Proof by Contradiction - Questions

Expand and simplify $x + y)^2$ .

Sarah claims that for any non-zero real number $k$ , the sum of $k$ and its reciprocal, $\frac{1}{k}$ , is always greater than or equal to 2.

Sarah tests two values: When $k = 3$ , $\frac{1}{3} = 3.\dot{3} \ge 2$ . When $k = 1$ , $\frac{1}{1} = 2 \ge 2$ .

Provide a counter-example to show that Sarah's claim is incorrect.

Given that $x$ and $y$ are distinct positive real numbers, use proof by contradiction to prove that:

\frac{x}{y} + \frac{y}{x} > 2

Expand and simplify $x + y)^2$ .

Sarah claims that for any non-zero real number $k$ , the sum of $k$ and its reciprocal, $\frac{1}{k}$ , is always greater than or equal to 2.

Sarah tests two values: When $k = 3$ , $\frac{1}{3} = 3.\dot{3} \ge 2$ . When $k = 1$ , $\frac{1}{1} = 2 \ge 2$ .

Provide a counter-example to show that Sarah's claim is incorrect.

Given that $x$ and $y$ are distinct positive real numbers, use proof by contradiction to prove that:

\frac{x}{y} + \frac{y}{x} > 2

1.1 Proof by Contradiction