5 Fundamental Properties of Inequalities Every Student Should Know
Understanding the Properties of Inequalities is essential in mathematics. This topic shows how to compare numbers and solve algebraic expressions that do not have an exact value. Mastering inequalities is important for exams such as CBSE, JEE, and other competitive tests, as well as for making logical comparisons and estimates in real-life scenarios.
What are Properties of Inequalities?
The properties of inequalities are a set of algebraic rules that explain how inequalities behave when numbers are added, subtracted, multiplied, or divided on both sides of an inequality. These properties ensure you can manipulate inequalities just like you do with equations, but with some unique rules—for example, the direction of the inequality sign may change when multiplying or dividing by a negative number.
Key Properties of Inequalities
| Property Name | Definition | Example |
|---|---|---|
| Transitive Property | If \(a < b\) and \(b < c\), then \(a < c\). | If 3 < 5 and 5 < 8, then 3 < 8. |
| Addition (Additive) Property | If \(a < b\), then \(a + c < b + c\). | If 2 < 7, then \(2 + 4 < 7 + 4\), i.e., 6 < 11. |
| Subtraction Property | If \(a < b\), then \(a - c < b - c\). | If 8 < 13, then \(8 - 3 < 13 - 3\), i.e., 5 < 10. |
| Multiplication (Multiplicative) Property | If \(a < b\), then \(ac < bc\) for \(c > 0\), but \(ac > bc\) for \(c < 0\). | If 2 < 3, then \(2 \times 4 < 3 \times 4\) (8 < 12); but \(2 \times -1 > 3 \times -1\) (-2 > -3). |
| Reflexive and Symmetric Properties | \(a = a\); If \(a < b\), then \(b > a\). | 7 = 7; If 2 < 5, then 5 > 2. |
Understanding How Inequality Properties Work
Let’s see each property in detail—with easy explanations and step-by-step examples to make the concepts clear.
1. Transitive Property
If \(a < b\) and \(b < c\), then \(a < c\). This allows you to connect multiple inequalities. Similarly, if \(a > b\) and \(b > c\), then \(a > c\).
Example: If you know Raman scored less than Priya, and Priya scored less than Ali, then Raman scored less than Ali.
2. Addition and Subtraction Properties
If you add or subtract the same number to both sides of an inequality, the inequality does not change.
- If \(a < b\), then \(a + c < b + c\) and \(a - c < b - c\).
Example: If 5 < 9, then 5 + 2 < 9 + 2 → 7 < 11; and 5 - 1 < 9 - 1 → 4 < 8.
3. Multiplication and Division Properties
- If \(a < b\) and you multiply or divide both sides by a positive number, the sign stays the same.
- If you multiply or divide both sides by a negative number, the sign reverses.
Example:
- Multiply by +3: If 2 < 5, then 2 × 3 < 5 × 3 → 6 < 15.
- Multiply by -2: If 2 < 5, then 2 × (-2) > 5 × (-2) → -4 > -10.
Always check the direction of the sign when dealing with negatives!
4. Reflexive and Symmetric Properties
Every number is equal to itself (reflexive). If \(a < b\), then \(b > a\) (symmetric).
Example: If 1 < 9, then 9 > 1.
Worked Examples (More examples here)
Example 1: Sign Flipping with Negatives
Solve: If -4 < 3, what happens if you multiply both sides by -1?
- Multiply both sides: (-4) × -1 = 4; 3 × -1 = -3.
- Flip the inequality: 4 > -3
Example 2: Addition Property
Given \(x - 2 > 7\), add 2 to both sides to keep the inequality balanced:
\(x - 2 + 2 > 7 + 2\) → \(x > 9\)
Example 3: Number Line Representation
Inequalities like \(x < 5\) can be plotted on the number line as all points to the left of 5 (not including 5).
Practice Problems
- If \(x - 3 < 5\), what is the solution for \(x\)?
- If \(a > b\), what is the result of multiplying both sides by -2?
- True or False: If 7 < 9, then 7 × (-1) < 9 × (-1).
- Solve: If \(2x + 5 < 9\), find the range for \(x\).
- If \(y > 3\), what happens when you add 6 to both sides?
Common Mistakes to Avoid
- Forgetting to flip the sign when multiplying or dividing both sides by a negative number.
- Treating inequality signs like equality signs in every situation—they are similar but not identical!
- Incorrectly adding or subtracting terms from only one side of the inequality.
- Using the wrong direction when rearranging inequalities.
Real-World Applications of Inequalities
Inequalities are used in real life for budgeting, estimating required resources, checking if a solution is within safety limits, and comparing measurements. For example, if a vehicle’s speed limit is less than 50 km/h, you can use inequalities to calculate the maximum time for a journey. In exams like JEE and NEET, questions often require you to solve, graph, or analyze inequalities as part of algebraic problem solving.
Related Topics and Further Study
- Rules of Inequality
- Properties of Equality
- Linear Inequalities in Two Variables
- Properties of Real Numbers
In this topic, we covered the properties of inequalities, key rules, common mistakes, and real-world uses. Mastering these principles helps you handle all types of comparison-based questions easily—whether in exams or daily life. Explore more on Vedantu to strengthen your algebra and problem-solving skills.
FAQs on Properties of Inequalities: Essential Rules, Tips & Practice
1. What are the 5 properties of inequality?
The five key properties of inequalities are: Reflexive Property (a ≤ a), Symmetric Property (If a ≤ b, then b ≥ a), Transitive Property (If a ≤ b and b ≤ c, then a ≤ c), Addition Property (If a ≤ b, then a + c ≤ b + c), and Multiplication Property (If a ≤ b and c > 0, then ac ≤ bc; if a ≤ b and c < 0, then ac ≥ bc). These properties are crucial for manipulating and solving inequalities in algebra.
2. What are the 4 rules of inequalities?
The four fundamental rules for working with inequalities are closely related to the properties above: You can add or subtract the same number from both sides; you can multiply or divide both sides by the same positive number; when multiplying or dividing by a negative number, you must reverse the inequality sign; and the transitive property allows you to combine inequalities if they share a common term.
3. What is inequality property?
An inequality property is a rule that governs how inequalities behave under different mathematical operations. These properties dictate how you can manipulate inequalities while maintaining their truth. Understanding these properties is essential for solving inequality problems and proving mathematical statements.
4. What happens if you multiply both sides of an inequality by a negative number?
When you multiply (or divide) both sides of an inequality by a negative number, you must reverse the direction of the inequality sign. For example, if x < 5, then multiplying by -1 gives -x > -5. This is because multiplying by a negative number changes the relative positions of the numbers on the number line.
5. How do you solve inequalities in algebra?
Solving algebraic inequalities involves isolating the variable using similar steps to solving equations. However, remember to reverse the inequality sign when multiplying or dividing by a negative number. Use the properties of inequalities to manipulate the equation, maintaining the integrity of the inequality. The solution will often be a range of values, rather than a single value.
6. What is the difference between the properties of equality and properties of inequality?
The main difference lies in how they behave with multiplication/division. Equality remains unchanged when multiplying/dividing by any non-zero number. However, inequalities change direction (the sign reverses) when multiplying/dividing by a negative number. Both share the reflexive, symmetric, and transitive properties.
7. Can inequalities be graphed on a number line?
Yes, inequalities can be effectively represented on a number line. For example, x > 2 would be shown as an open circle at 2 and an arrow extending to the right. x ≤ 2 would use a closed circle at 2 and an arrow to the left. Number line graphs visually show the solution set of an inequality.
8. Why is flipping the inequality sign required when multiplying/dividing by a negative number?
Flipping the inequality sign when multiplying or dividing by a negative number is necessary to maintain the truth of the inequality. Multiplying or dividing by a negative number reverses the order of numbers on the number line, thus requiring the inequality sign to be reversed to keep the statement accurate.
9. Properties of inequalities class 9?
In Class 9, you typically learn the basic properties of inequalities, including the addition, subtraction, multiplication, and division properties, as well as the transitive property. These are fundamental for solving linear inequalities and understanding number relationships.
10. Properties of inequalities of real numbers?
The properties of inequalities apply to real numbers. They define how the relations <, >, ≤, and ≥ behave under various operations. The rules remain consistent across different subsets of real numbers, such as integers or rational numbers.
