13
Basic properties of numerical inequalities
Textbook: pp. 71–74
GoalKnow and apply the basic properties of numerical inequalities (transitivity, adding, multiplying).
New words
transitivity · tranzitivlik xossasiinequality sign · tengsizlik ishorasiadding to both sides · ikkala qismiga qo‘shishsign reversal · ishora o‘zgarishi
Explanation
Property 1: if a > b and b > c then a > c. Property 2: if the same number is added to both sides of an inequality, the sign does not change; hence a term can be moved to the other side with its sign changed. Property 3: if both sides are multiplied (or divided) by the same positive number, the sign stays; if by the same negative number, the sign reverses. The proof rests on the sign of the difference: if a > b and c < 0, then (a - b)c < 0, that is ac < bc. These properties are used most often when solving inequalities.
Worked examples
Subtract 7 from both sides of 5 > 2: 5 - 7 > 2 - 7, that is -2 > -5.
Multiply 3 < 4 by -2: -6 > -8 (the sign reversed). Divide 12 > 8 by -4: -3 < -2.
Class activity
“Sign control”: the teacher names an inequality and an operation (for example, “multiply by -3”) and pupils raise a card with “>” or “<”.
Practice
1
If x > 3, compare x - 5 and -2.
x - 5 > -2
2
If a > b, write the relation between -a and -b.
-a < -b
3
From -3x > 12 find x.
x < -4
4
Why does the inequality sign reverse when multiplying by a negative number?
Because a - b > 0 and c < 0 give (a - b)c < 0, so ac - bc < 0 and ac < bc.