Properties of Inequality

Inequality properties describe how inequality relationships behave when we manipulate expressions. These rules are essential for solving and analyzing inequalities correctly and consistently.

Logical and Operational Properties

1. Trichotomy Property

$\forall a, b \in \mathbb{R},\ a < b \ \lor \ a = b \ \lor \ a > b$

This is read as, for any two numbers a and b from the real number set, either a is less than b, or a is equal to b, or a is greater than b. This is a basic property of the number line that while comparing a number from the first, either the number is behind it, on top of it, or ahead of it.

2. Transitive Property

$\forall a, b, c \in \mathbb{R},\ a < b \land b < c \Rightarrow a < c$

This is read as, for all a, b, and c in the real number set, if $a < b$ and $b < c$ , then it implies that $a < c$ . This is true vice versa for greater than as well written as;

$\forall a, b, c \in \mathbb{R},\ a > b \land b > c \Rightarrow a > c$

3. Additive Property

$\forall a, b, c \in \mathbb{R},\ a < b \Rightarrow a + c < b + c$

$\forall a, b, c \in \mathbb{R},\ a > b \Rightarrow a + c > b + c$

This property explains that given an inequality expression, we can add the same number to both sides of the inequality and the inequality sign will be preserved.

4. Multiplicative Property

$\forall a, b, c \in \mathbb{R},\ c > 0 \land a < b \Rightarrow ac < bc$

$\forall a, b, c \in \mathbb{R},\ c > 0 \land a > b \Rightarrow ac > bc$

When both sides of an inequality are multiplied by a positive number, the inequality direction remains unchanged.

$\forall a, b, c \in \mathbb{R},\ c < 0 \land a < b \Rightarrow ac > bc$

$\forall a, b, c \in \mathbb{R},\ c < 0 \land a > b \Rightarrow ac < bc$

When both sides of an inequality are multiplied by a negative number, the inequality direction flips.

Combined Inequality Operations

Addition of Inequalities

$\forall a, b, c, d \in \mathbb{R},\ a > b \land c > d \Rightarrow a + c > b + d$

$\forall a, b, c, d \in \mathbb{R},\ a < b \land c < d \Rightarrow a + c < b + d$

If two inequalities point the same way, you can add their left and right sides to get a new true inequality.
Natural language: “Bigger plus bigger is still bigger; smaller plus smaller is still smaller.”

Multiplication of Positive Inequalities

$\forall a, b, c, d \in \mathbb{R},\ a > b > 0 \land c > d > 0 \Rightarrow ac > bd$

$\forall a, b, c, d \in \mathbb{R},\ 0 < a < b \land 0 < c < d \Rightarrow ac < bd$

If all values are positive and both pairs follow the same inequality direction, the inequality holds when multiplying the pairs.
Natural language: “A bigger positive times a bigger positive gives a bigger result.”

Activity: Match The Columns!

Reorder the Definition to match them with the correct Inequality Property.

Inequality Property

Definition

$\text{Trichotomy Property}$

$\text{Transitive Property}$

$\text{Additive Property}$

$\text{Multiplicative Property (Positive c)}$

True or False Activity!!!

Read each statement carefully and decide if it is true or false.

For any two real numbers a and b, exactly one of a $<$ b, a = b, or a $>$ b is true.

If a $<$ b and b $<$ c, then a $>$ c.

If a $<$ b, then a + c $<$ b + c.

If a $<$ b and you multiply both sides by a negative number, the inequality direction stays the same.

If a $<$ b and c $>$ 0, then ac $<$ bc.

Build the Mathematical Statements

Build the Correct Equations Activity

Trichotomy Property

Transitive Property of Inequality

Additive Property of Inequality

Multiplicative Property of Inequality

End of Lesson