Inequality

Properties

Multiplicative

Multiplying by a positive integer preserves the inequality $(a\le b)\wedge (c>0)⟹ac\le bc$ Multiplying by a negative integer triggers Negative Flip $(a<b)\wedge (c<0)⟹ac>bc$

Multiplicative Inverse

If both are positive, you can take the reciprocal and then flip the inequality $0<a<b⟹\frac{1}{a}>\frac{1}{b}$ If both are negative, you can take the reciprocal and then flip inequality $a<b<0⟹\frac{1}{a}>\frac{1}{b}$

Additive

Adding a number to both sides preserves the inequality $a<b⟹a+c<b+c$

$(a\le b)\wedge (c\le d)⟹a+c\le b+d$ Two different inequalities, combined will be another inequality