Inequalities are a fundamental part of mathematics used in everything from algebraic equations to real-world problem solving. Whether you’re a student, tutor, or professional, solving inequalities correctly is crucial for making informed decisions in academics, engineering, economics, or logic-based problems.
What is an Inequality?
An inequality is a mathematical statement that compares two expressions using inequality symbols:
- < (less than)
- ≤ (less than or equal to)
- > (greater than)
- ≥ (greater than or equal to)
- ≠ (not equal to)
Instead of finding a single number (as in equations), you find a range or set of values that satisfy the inequality.
What is the Inequality Calculator?
The Inequality Calculator is a math tool that allows users to solve algebraic inequalities online. It supports:
- Linear inequalities (e.g., 2x - 3 < 5)
- Quadratic inequalities (e.g., x² - 5x + 6 ≥ 0)
- Rational inequalities (e.g., (x+1)/(x−3) ≤ 0)
- Compound inequalities (e.g., -2 < x + 1 ≤ 5)
It provides detailed solutions that include steps like simplifying expressions, solving equations, analyzing sign charts, and expressing the final result in interval notation or number line format.
How to Use the Inequality Calculator
Using the calculator is easy and requires just a few steps:
- Enter the inequality expression into the input box.
Example:2x + 5 < 15 - Click the “Calculate” or “Solve” button.
- View the step-by-step solution, which includes:
- Simplification steps
- Critical points (roots)
- Sign chart analysis (for quadratic or rational cases)
- Final solution in set-builder or interval notation
Types of Inequalities the Calculator Solves
1. Linear Inequality
Example:3x - 7 > 5
Solution:x > 4
2. Quadratic Inequality
Example:x² - 3x - 4 ≤ 0
Factor: (x - 4)(x + 1) ≤ 0
Solution:x ∈ [-1, 4]
3. Rational Inequality
Example:(x - 2)/(x + 1) ≥ 0
Solution involves sign chart and finding where the expression is non-negative.
Result:x ∈ (-∞, -1) ∪ [2, ∞)
4. Compound Inequality
Example:-3 < 2x - 1 ≤ 7
Step 1: Solve both sides
Step 2: Final result:1 < x ≤ 4
Formula and Rules Used in Solving Inequalities
Here are the basic rules and formulas applied during inequality solving:
For Linear Inequalities:
To solve ax + b < c, isolate x:
cssCopyEditax < c - b x < (c - b)/a Rule:
When multiplying or dividing both sides of an inequality by a negative number, reverse the inequality sign.
Example:-2x > 6 → x < -3
For Quadratic Inequalities:
Given: ax² + bx + c < 0
- Find roots by factoring or quadratic formula
- Divide number line into intervals based on roots
- Test sign of expression in each interval
- Select intervals where the inequality is true
Quadratic formula:
iniCopyEditx = [-b ± √(b² - 4ac)] / 2a For Rational Inequalities:
Steps include:
- Set numerator and denominator to zero to find critical points
- Build a sign chart
- Determine positive/negative intervals
- Include or exclude endpoints based on inequality sign
Why Use an Inequality Calculator?
- ✅ Instant Results for homework, exams, and professional tasks
- ✅ Supports multiple types of inequalities
- ✅ Provides step-by-step explanations
- ✅ Handles complex expressions and compound inequalities
- ✅ Useful for high school and college-level math
Benefits of the Inequality Calculator
- No more manual errors
- Helps you understand each solving step
- Prepares students for exams like SAT, GRE, or algebra finals
- Ideal for math tutors and online educators
- Works on mobile and desktop devices
20 Frequently Asked Questions (FAQs)
1. What is an inequality?
A statement showing the relationship between two values using symbols like <, >, ≤, ≥.
2. Can the calculator solve compound inequalities?
Yes, it solves compound inequalities like a < x ≤ b.
3. Does the tool show steps?
Yes, it displays all intermediate steps used to reach the final answer.
4. What are linear inequalities?
Inequalities involving variables with power 1 (e.g., 2x + 3 < 10).
5. What is a quadratic inequality?
An inequality where the variable is squared, like x² - 4 ≥ 0.
6. Can it solve inequalities with fractions?
Yes, rational inequalities like (x-2)/(x+3) > 0 are supported.
7. Does it reverse the inequality when dividing by a negative number?
Yes, it applies the correct rule for sign reversal.
8. What is interval notation?
A way to express solution sets using brackets and parentheses (e.g., (2, ∞)).
9. Can it plot on a number line?
Some versions display solutions graphically on a number line.
10. Can I input absolute value inequalities?
Some advanced versions support it, but check if your tool allows |x - a| < b.
11. Can I solve 3-variable inequalities?
This calculator works best with single-variable inequalities.
12. How do I enter compound inequalities?
Enter in the format: a < expression < b.
13. Does it work with decimals?
Yes, you can enter decimal numbers.
14. Can I solve inequalities with square roots?
Yes, as long as it's algebraically solvable.
15. Is it useful for test prep?
Absolutely. It helps students understand solution methods quickly.
16. Is it free to use?
Yes, the inequality calculator is 100% free.
17. Do I need to install anything?
No. It works directly in your browser.
18. Does it support mobile devices?
Yes, the tool is mobile-responsive.
19. Can teachers use it for demonstrations?
Yes, it's excellent for math instruction.
20. Is it better than solving manually?
It’s ideal for checking work or learning step-by-step processes.
Final Thoughts
The Inequality Calculator is a powerful educational and problem-solving tool. It offers a quick and accurate way to solve linear, quadratic, and compound inequalities with ease. From high school algebra students to professionals tackling logical constraints, this tool saves time and enhances understanding.