2.1: Properties of Inequalities
Here are some important properties of inequalities:
If \(a\), \(b\), and \(c\) are real numbers, then:
Transitive Property if \(a < b\) and \(b < c\) then \(a < c\)
Addition Property if \(a < b\) then \(a + c < b + c\)
Subtraction Property if \(a < b\) then \(a − c < b − c\)
Multiplication Property (Multiplying by a positive number) if \(a < b\) and \(c > 0\) then \(ac < bc\)
Multiplication Property (Multiplying by a negative number) if \(a < b\) and \(c < 0\) then \(ac > bc\)
Division Property (Dividing by a positive number) if \(a < b\) and \(c > 0\) then \(\dfrac{a}{c} < \dfrac{b}{c}\)
Division Property (Dividing by a negative number) if \(a < b\) and \(c < 0\) then \(\dfrac{a}{c} > \dfrac{b}{c}\)
Transitive Property
If 3 < 7 and 7 < 14 then...
Solution
3 < 14
Addition Property
If 3 < 7, then add 4 to both sides.
Solution
3 + 4 < 7 + 4
7 < 11
Subtraction Property
If 3 < 7, subtract 6 on both sides
Solution
3 < 7
3 - 6 < 7 - 6
-3 < 1
Multiplication Property (Multiplying by a positive number)
If 3 < 7, multiply both sides by 5.
Solution
3 < 7
3 * 5 < 5 * 7
15 < 35
Multiplication Property (Multiplying by a negative number)
If 3 < 7, multiply both sides by -4.
Solution
3 < 7
3 * -4 ? -4 * 7
-12 ? -28
-12 > -28 The direction of the inequality is changed.
Division Property (Dividing by a positive number)
If 6 < 8, divide both sides by 2.
Solution
6/2 < 8/2
3 < 4
Division Property (Dividing by a negative number)
If 9 < 15, divide both sides by -3.
Solution
9 < 15
9/-3 ? 15/-3
-3 ? -5
-3 > -5
The direction of the inequality is changed.