Given that √3 is irrational, prove that 5 + 2√3 is irrational.

How to lay out the answer. Follow the frame, then write it in your own words.

1 · Write what is given and what is to be found. Given: √3 is irrational. To prove: 5 + 2√3 is irrational.
2 · Write the fact used. If a and b are integers and b ≠ 0, then a/b is rational. Sums, differences and products of integers are integers.
3 · Assume the opposite. Assume 5 + 2√3 is rational. Then 5 + 2√3 = a/b, where a and b are co-prime integers and b ≠ 0.
4 · Isolate √3. 2√3 = a/b − 5 = (a − 5b)/b, so √3 = (a − 5b)/(2b).
5 · Reach the contradiction. (a − 5b)/(2b) is rational, so √3 would be rational. This contradicts the given fact that √3 is irrational.
6 · Check and box the answer. So our assumption is wrong. Therefore 5 + 2√3 is irrational.

Scoring tip: Show the line √3 = (a − 5b)/(2b) clearly. It is the step that carries the method mark.