12
Numerical inequalities
Textbook: pp. 68–70
GoalKnow the definition of a numerical inequality; compare two numbers by their difference and prove simple inequalities.
New words
numerical inequality · sonli tengsizlikcomparing numbers · sonlarni taqqoslashsign of the difference · ayirmaning ishorasinumber line · son o‘qi
Explanation
A number a is greater than b (a > b) if the difference a - b is positive; a < b means a - b is negative. For any two numbers a and b exactly one of a > b, a = b, a < b holds. To compare numbers means to decide which of the signs >, = or < fits between them; we find the sign of the difference. On the number line the greater number lies to the right of the smaller one. The sign of a difference can also prove inequalities: since a² + b² - 2ab = (a - b)² ≥ 0, the inequality a² + b² ≥ 2ab is always true.
Worked examples
7/9 and 3/4: 7/9 - 3/4 = (28 - 27)/36 = 1/36 > 0, so 7/9 > 3/4.
Prove a² + b² ≥ 2ab: a² + b² - 2ab = (a - b)² ≥ 0. Equality holds only when a = b.
Class activity
“Find the greater”: cards hold fractions, squares and roots; a pair compares two of them using the difference and shows the result on a number line.
Practice
1
Compare -0.35 and -3/8.
-0.35 > -3/8
2
Which is greater, (x + 2)² or x² + 4x + 3?
(x + 2)² is greater: the difference is 1 > 0.
3
If a - b = -5, which is greater, a or b?
b is greater
4
Why is a² + b² ≥ 2ab true for all a and b?
Because the difference is the square (a - b)², and a square is never negative.