Transitive
Property of Inequalities
Any of the following properties:
If a < b and b < c ,
then a < c.
If a ≤ b and b ≤ c , then a ≤ c.
If a > b and b > c , then a > c.
If a ≥ b and b ≥ c , then a ≥ c.
Note: This is a property of equality and inequalities. (Click here for the transitive property of equality.) One must be cautious, however, when attempting to develop arguments
using the transitive property in
other settings.
Here is an example of an unsound application
of the transitive property: "Team A defeated team B, and
team B defeated team C. Therefore, team A will defeat team C."
See
also
Transitive
property of equality, reflexive
property of equality, symmetric
property of equality, inequality
rules
Worked Example
Problem: Suppose you know that x < 7 and 7 < 12. What can you conclude about x and 12?
Step 1: Identify the three quantities and the shared middle value. Here a = x, b = 7, and c = 12.
x<7and7<12 Step 2: Both inequalities point in the same direction (both use <), and the middle value 7 appears on the right of the first inequality and the left of the second. The transitive property applies directly.
Step 3: Apply the transitive property: since x < 7 and 7 < 12, conclude x < 12.
x<7<12⟹x<12 Answer: By the transitive property of inequalities, x < 12.
Another Example
Problem: You are told that a ≥ b and b ≥ 10. What can you say about a and 10?
Step 1: Identify the chain: a ≥ b and b ≥ 10. The shared link is b.
a≥bandb≥10 Step 2: Both inequalities use ≥ and chain through b, so the transitive property for ≥ applies.
Step 3: Conclude that a ≥ 10.
a≥b≥10⟹a≥10 Answer: By the transitive property, a ≥ 10.
Frequently Asked Questions
Does the transitive property work if the inequality signs point in different directions?
No. Both inequalities must point in the same direction for the transitive property to apply. If you have a < b and b > c, you cannot conclude anything about how a and c compare. You would need to rearrange or combine the inequalities another way.
Can you mix strict and non-strict inequalities with the transitive property?
Yes, with care. If a < b and b ≤ c, you can still conclude a < c (the result keeps the strict inequality). Likewise, if a ≤ b and b < c, then a < c. The key rule is: if at least one of the two inequalities is strict, the conclusion is strict.
Transitive Property of Inequalities vs. Transitive Property of Equality
The equality version says if a = b and b = c, then a = c. The inequality version replaces '=' with an inequality symbol (<, ≤, >, or ≥). Both properties let you 'chain' two statements through a shared middle value to reach a conclusion about the outer values. The difference is simply which relation — equality or inequality — is being chained.
Why It Matters
The transitive property of inequalities is essential whenever you solve multi-step inequalities or chain comparisons. For instance, when solving a system of inequalities you often deduce that one variable is bounded by a second, and that second is bounded by a third; transitivity lets you skip the middle term and state the final bound directly. It also underpins proofs in algebra and analysis where you establish ordering among several expressions.
Common Mistakes
Mistake: Applying transitivity when the inequality signs point in opposite directions, such as concluding a < c from a < b and b > c.
Correction: The transitive property requires both inequalities to have the same direction. If the signs differ, you cannot draw a direct conclusion about a and c without additional information.
Mistake: Losing the strict inequality when mixing < with ≤. For example, writing a ≤ c when a < b and b ≤ c.
Correction: When at least one of the two inequalities is strict (< or >), the conclusion is also strict. So a < b and b ≤ c gives a < c, not merely a ≤ c.
