Numerical inequalities and their properties
Compare numbers by the sign of the difference a − b, prove inequalities, add inequalities term by term and estimate the value of an expression.
- Ukraine: grade 9
- Russia: grade 8
Guide
In Ukrainian Grade 9 algebra (§1–3) numerical inequalities start from the difference: a is greater than b if the difference a − b is positive, and less than b if it is negative.
So to compare two numbers you find the sign of their difference. Inequalities are proved the same way: change the difference until its sign is plain to see.
Theorems: if a > b and b > c then a > c. If a > b then a + c > b + c. If a > b and c > 0 then ac > bc, but if c < 0 then ac < bc.
The inequalities a > b and c > d have the same sign. You can add them term by term: a + c > b + d. If all the numbers are positive you can multiply them term by term too: ac > bd. This is how the value of an expression is estimated.