Inequalities: Rules, Worked Examples, and Practice Exercises
Inequalities: Rules, Examples, and Exercises
Inequalities are mathematical tools that allow us to compare quantities and determine which values satisfy a condition. They are used to express order relationships, establish limits, solve age problems, calculate budgets, and analyze situations where a quantity must be greater, less than, or different from another.
Understanding this topic requires distinguishing between a numerical inequality, an algebraic inequality, and an inequation. It is also important to learn how to transform expressions without altering the solution set. One rule deserves special attention: when multiplying or dividing both sides of an inequality by a negative number, the direction of the sign must be reversed.
In this guide, you will study inequality symbols, their properties, solving first-degree inequalities, representing solutions on the number line, and the most common errors. The examples are developed step by step so you can check each operation and understand what the result means.
In Summary
An inequality compares two expressions using signs such as <, >, ≤, or ≥. An inequation is an inequality containing one or more unknowns whose solution consists of finding the values that make it true.
1. What are inequalities?
The mathematical inequality
An inequality states that two quantities or expressions do not necessarily have the same value, expressing an order relationship between them. For example, 3 < 7 indicates that three is less than seven. This statement is true because the quantity on the left is smaller than the one on the right.
An inequality can compare concrete numbers or algebraic expressions. For example, x + 2 > 5 expresses a condition whose fulfillment depends on the value assigned to the variable x.
The inequation
An inequation is a mathematical inequality containing an unknown and stating a condition that must be met. Solving it means determining all values of the unknown that make the inequality true, considering the established domain.
Numerical inequality example
It is a true statement and does not require finding an unknown.
Inequation example
It is a condition that is met when x is greater than 3.
Fundamental idea: every inequation is expressed through an inequality, but not every inequality contains an unknown that needs to be solved.
Inequality symbols
| Symbol | Meaning | Example |
|---|---|---|
| < | Less than | 2 < 6 |
| > | Greater than | 9 > 4 |
| ≤ | Less than or equal to | x ≤ 5 |
| ≥ | Greater than or equal to | x ≥ 2 |
| ≠ | Not equal to | x ≠ 0 |
The signs < and > express strict inequalities: they do not include equality. On the other hand, ≤ and ≥ indicate that equality is also permitted. The symbol ≠ expresses that two quantities are different, but by itself does not establish which one is greater.
2. Properties and rules for solving inequalities
The properties of inequalities allow transforming an inequality into an equivalent one—that is, one that preserves the exact same solution set. The key is to correctly apply the same operation to both sides and respect the conditions of each rule.
Rule 1. Adding or subtracting the same quantity
When adding or subtracting the same number on both sides, the direction of the inequality is preserved.
For example, if x + 4 < 10, we subtract 4 from both sides:
Original expression: x + 4 < 10
Operation: subtract 4 from both sides.
x + 4 − 4 < 10 − 4
Simplification: x < 6
Solution: all numbers less than 6.
Rule 2. Multiplying or dividing by a positive number
When both sides are multiplied or divided by the same positive number, the inequality sign is preserved.
For example, to solve 3x > 12, we divide both sides by 3:
3x > 12
3x ÷ 3 > 12 ÷ 3
Result: x > 4
Solution: x > 4.
Rule 3. Multiplying or dividing by a negative number
When both sides are multiplied or divided by a negative number, the direction of the inequality sign is reversed. This rule is essential for correctly solving inequalities.
For example, let's solve −2x < 8.
Given: −2x < 8.
Approach: isolate the unknown x.
Rule: divide both sides by −2 and reverse the sign.
−2x ÷ (−2) > 8 ÷ (−2)
Operation: x > −4.
Result: x > −4.
Interpretation: any number greater than −4 satisfies the original inequality.
Warning: the sign is not reversed simply because a negative number appears in the expression. It is reversed when both sides are multiplied or divided by a negative quantity.
Rule 4. The transitive property
If one quantity is less than a second, and the second is less than a third, then the first is also less than the third.
For example, if 2 < 5 and 5 < 9, then 2 < 9.
3. How to solve a first-degree inequality step by step
A first-degree inequality with one unknown can be written in its general form as ax + b < c, or using any of the other inequality signs, where a, b, and c are real numbers and a is non-zero. To solve it, equivalent operations are applied until the unknown is isolated.
The general procedure consists of simplifying both sides, gathering the terms with the unknown, transferring constants using equivalent operations, and dividing by the variable's coefficient. If this coefficient is negative, the sign must be reversed in the final step.
Solved example: 2x + 3 < 11
GIVEN
APPROACH
Determine the values of x that make the inequality true.
RULE OR FORMULA
Subtract the same quantity from both sides and then divide by a positive number, keeping the sign.
STEP 1. Subtract 3 from both sides
STEP 2. Simplify
STEP 3. Divide by 2
RESULT
INTERPRETATION
The solution comprises all real numbers less than 4. The number 4 is not included because the original inequality is strict.
Checking the result
One way to check a solution is to substitute a value that belongs to the obtained set and another that does not belong to it.
Check with x = 3
2(3) + 3 < 11
9 < 11
True: 3 belongs to the solution.
Check with x = 5
2(5) + 3 < 11
13 < 11
False: 5 does not belong to the solution.
Solved example with negative coefficient: −3x + 6 ≥ 15
GIVEN
STEP 1. Subtract 6 from both sides
STEP 2. Divide by −3
Since the divisor is negative, the sign ≥ is reversed and becomes ≤.
RESULT
INTERPRETATION
The solution includes all real numbers less than or equal to −3.
Check with x = −3
−3(−3) + 6 ≥ 15
15 ≥ 15
True. The endpoint −3 does belong to the solution.
4. How to represent solutions on the number line
The number line allows visualizing the set of values that satisfies an inequality. In strict inequalities, the endpoint does not belong to the solution set and is represented with an open circle. In inequalities that include equality, the endpoint does belong to the set and is represented with a closed circle.
Solution: x < 4
x < 4 · Open circle at 4
4 is not included. The turquoise stripe to the left represents all numbers less than 4.
The open circle indicates that 4 is not included. The arrow pointing left represents numbers less than 4. The solution also extends to values less than zero; it does not start at 0.
Solution: x ≤ −3
x ≤ −3 · Closed point at −3
−3 is included. The turquoise stripe to the left represents −3 and all smaller numbers.
The closed circle indicates that −3 belongs to the solution. The arrow pointing left represents numbers less than −3 and −3 itself.
Interval notation
Solutions can also be expressed using intervals. In standard notation, parentheses indicate that an endpoint is not included, and brackets indicate that it is included. Parentheses are always used with infinity, because infinity is not a real number that can belong to the interval.
| Inequality | Interval notation | Endpoint |
|---|---|---|
| x < 4 | (−∞, 4) | 4 is not included |
| x ≤ 4 | (−∞, 4] | 4 is included |
| x > −3 | (−3, +∞) | −3 is not included |
| x ≤ −3 | (−∞, −3] | −3 is included |
5. Common errors and applications
Common errors when solving inequalities
Error 1. Not reversing the sign when dividing by a negative number.
Incorrect: −2x < 8 → x < −4.
Correct: −2x < 8 → x > −4.
Error 2. Confusing < with ≤.
The expression x < 4 excludes 4; the expression x ≤ 4 includes it.
Error 3. Giving a single value as a solution.
In an inequality like x < 4, the solution is not solely 3 or 2. It is the set of all real numbers less than 4.
Error 4. Incorrectly representing the endpoint.
On the number line, strict inequalities use an open circle and inequalities with equality use a closed circle.
Error 5. Applying operations without considering their conditions.
If you multiply or divide by an expression containing the unknown, it is necessary to study its sign and the values for which the operation is defined. You cannot mechanically apply the positive or negative number rule without verifying these conditions.
Applications of inequalities
Inequalities allow modeling situations where a quantity must remain below a limit, reach a minimum, or fall within an allowed interval. Interpretation depends on the context: a mathematical solution may require additional restrictions when it represents countable objects, money, ages, or physical magnitudes.
Applied example: maximum budget
A person has $500 to buy notebooks that cost $40 each. If x represents the number of notebooks and other expenses are not considered, the cost must not exceed the available budget.
GIVEN
Budget: $500. Price per notebook: $40.
APPROACH
OPERATION
MATHEMATICAL RESULT
If x is a real number, the solution is x ≤ 12.5. However, the number of notebooks must be a non-negative integer.
CONTEXTUAL INTERPRETATION
A maximum of 12 notebooks can be bought without exceeding the budget. Twelve notebooks cost $480; thirteen cost $520, which is greater than $500.
This example shows why it is necessary to distinguish between algebraic solution and contextual interpretation. The former is obtained through mathematical operations; the latter considers the real conditions of the problem.
6. Frequently asked questions about inequalities
What is the difference between an inequality and an inequation?
An inequality expresses an order relationship between two quantities or expressions. An inequation is an inequality containing an unknown whose solution requires finding the values that satisfy the condition.
What are the main inequality symbols?
The main signs are < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). The symbol ≠ means not equal to, although it does not indicate an order relationship by itself.
When is the direction of the inequality sign changed?
When both sides are multiplied or divided by a negative number. For example, −2x < 8 transforms into x > −4 when divided by −2.
How is x < 3 represented on the number line?
An open circle is drawn at 3 and the region to the left is highlighted, because the solution comprises all real numbers less than 3, excluding the endpoint.
What does x ≥ 5 mean?
It means that x can take any real value greater than 5 or be exactly equal to 5. On the number line, a closed circle is used at 5 and the region to the right is highlighted.
Can an inequality have infinite solutions?
Yes. For example, x < 4 has infinite real solutions, because there are infinite real numbers less than 4.
How do you check the solution of an inequality?
You can substitute values belonging to the solution set and verify that they satisfy the original inequality. Algebraic transformations and interval limits can also be checked.
What happens if an inequality has no solution?
If the conditions are incompatible, no value in the considered domain satisfies them. For example, x + 2 < x has no solution in real numbers, because subtracting x from both sides yields 2 < 0, a false statement.
7. Practice exercises
Solve the following inequalities without consulting the answers first. Remember to justify your operations and pay special attention to negative coefficients.
Exercise 1. x + 7 > 12
Exercise 2. 4x ≤ 20
Exercise 3. −5x > 15
Exercise 4. 3x − 4 ≥ 8
Exercise 5. 2x + 5 < 17
Exercise 6. −2x + 1 ≤ 9
Answers to check your learning
1. x + 7 > 12 → x > 5
2. 4x ≤ 20 → x ≤ 5
3. −5x > 15 → x < −3
4. 3x − 4 ≥ 8 → x ≥ 4
5. 2x + 5 < 17 → x < 6
6. −2x + 1 ≤ 9 → x ≥ −4
Conclusion
Inequalities allow expressing order relationships, establishing limits, and solving mathematical and everyday problems. Mastering them requires understanding the meaning of the signs, correctly applying properties, and representing solutions as sets of values rather than as isolated numbers.
The most important principle is to preserve the equivalence of each transformation. Adding or subtracting the same quantity on both sides does not change the direction of the inequality; neither does multiplying or dividing by a positive number. In contrast, multiplying or dividing by a negative number forces the sign to be reversed.
Learning to check results and interpret solutions in context helps develop more precise algebraic reasoning. Before finishing an inequality, verify the operations, the direction of the sign, the endpoints, and the problem's constraints.
Continue learning with Aprende Professional Group | Learning Club.
Constant practice and reasoned verification of each procedure are fundamental to building a solid understanding of algebra.
Continue learning
- Algebraic language: expressions, variables, and statement translation.
- Operations with monomials, binomials, trinomials, and polynomials.
- First-degree equations: procedure and solved exercises.
Educational material for middle and high school students. It is recommended to solve the exercises in writing and check each answer by substitution.
Keep learning with Frecuencia Educativa
This topic is part of the following episode:
Season 2, Episode 20 | Inequalities and Inequations | Frecuencia Educativa
In this article, we take a deep dive into the subject of inequalities and inequations.
Listen to the full episode here:
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Watch this short video, where we explain how to correctly apply inequalities and inequations.
Classification Record
Field | Classification |
|---|---|
Collection | Mathematics |
Center | Mathematics |
Area | Algebra |
Topic | Inequalities and Inequations |
Resource Type | Master Article |
Educational Level | Middle School and High School |
Language | English |
References
1. OpenStax. (n.d.). 2.7 Solve linear inequalities. In Elementary Algebra 2e. Rice University.
https://openstax.org/books/elementary-algebra-2e/pages/2-7-solve-linear-inequalities
2. OpenStax. (n.d.). Elementary Algebra 2e. Rice University.
https://openstax.org/details/books/elementary-algebra-2e
3. Khan Academy. (n.d.). Inequalities.
https://www.khanacademy.org/math/algebra
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