For a>b>0 and m>0, which inequality always holds?

01 / Standard#Your turn

Official paper · jm01-2022 · I.2 · PDF 2

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

The denominators a and a+m are positive.

Hint 2

Subtract b/a from (b+m)/(a+m).

Worked solution

Put the difference over a common denominator.
$\frac{b+m}{a+m}-\frac ba=\frac{m(a-b)}{a(a+m)}$
Every factor has the required positive sign.
$m(a-b)>0,\quad a(a+m)>0\implies\frac ba<\frac{b+m}{a+m}$

Checks and common pitfalls: Expressions involving b−m can change sign or be undefined.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Put the difference over a common denominator.
$\frac{b+m}{a+m}-\frac ba=\frac{m(a-b)}{a(a+m)}$
Every factor has the required positive sign.
$m(a-b)>0,\quad a(a+m)>0\implies\frac ba<\frac{b+m}{a+m}$

Think first. Reveal a hint when the class is ready.

2022 JM01