Solve and understand inequalities with our intuitive step-by-step calculator.
Inequality Solver
11 or x^2 – 4x + 3
Use 'x' as the variable. For quadratic, use '^' for powers (e.g., x^2). Operators: , =, =.
Real Numbers
Integers
Select if you are looking for real number solutions or only integer solutions.
Results
—
Interval Notation: —
Set Notation: —
Number of Integer Solutions: —
Formula Used: The calculator parses the inequality, identifies its type (linear or quadratic), and applies algebraic manipulation to isolate the variable. For quadratic inequalities, it finds roots and tests intervals. Solutions are presented in interval and set notation.
Visual representation of the inequality solution set.
Step
Action
Resulting Inequality
Enter an inequality and click Calculate.
Step-by-step breakdown of the inequality solution process.
What is an Inequality Calculator Step by Step?
An inequality calculator step by step is a powerful online tool designed to help users solve mathematical inequalities. Unlike a simple calculator that might just give a final answer, this type of tool breaks down the solution process into a series of logical, easy-to-follow steps. This is crucial for understanding the underlying mathematical principles and for learning how to solve inequalities manually. It typically handles various types of inequalities, including linear, quadratic, and sometimes even polynomial or rational inequalities.
Who should use it?
Students: High school and college students learning algebra and pre-calculus will find it invaluable for homework, studying, and exam preparation. It provides immediate feedback and clarifies complex procedures.
Educators: Teachers can use it to demonstrate the solving process in class, create examples, or assign problems that require understanding the steps.
Anyone needing to solve inequalities: Whether for academic purposes or practical applications in fields like engineering, economics, or computer science, this tool offers a reliable way to find solutions.
Common Misconceptions:
"It just gives the answer": A good step-by-step calculator explains *how* it gets the answer, not just what the answer is.
"All inequalities are solved the same way": Different types of inequalities (linear vs. quadratic) require different methods. The calculator should differentiate.
"The direction of the inequality sign never changes": Multiplying or dividing by a negative number reverses the inequality sign. This is a common pitfall the calculator should highlight.
Inequality Calculator Step by Step Formula and Mathematical Explanation
The core of an inequality calculator step by step lies in its ability to apply fundamental algebraic rules to manipulate inequalities. The process varies depending on the type of inequality.
Linear Inequalities (e.g., ax + b > c)
The goal is to isolate the variable 'x'. This involves performing inverse operations, similar to solving linear equations, with one critical difference: if you multiply or divide both sides by a negative number, you must reverse the inequality sign.
Steps:
Simplify both sides of the inequality if necessary.
Move all terms containing the variable to one side and all constant terms to the other.
Combine like terms.
If the variable's coefficient is not 1, divide both sides by the coefficient. Remember to reverse the inequality sign if the coefficient is negative.
Example Derivation: Solve 3x - 7 < 8
Add 7 to both sides: 3x - 7 + 7 < 8 + 7 => 3x < 15
Divide both sides by 3 (a positive number, so the sign stays the same): 3x / 3 < 15 / 3 => x < 5
Quadratic Inequalities (e.g., ax^2 + bx + c > 0)
Solving quadratic inequalities involves finding the roots (where the expression equals zero) and then testing intervals on a number line.
Steps:
Rewrite the inequality so that one side is 0 (e.g., ax^2 + bx + c > 0).
Find the roots of the corresponding quadratic equation ax^2 + bx + c = 0 using factoring, the quadratic formula, or completing the square.
Plot these roots on a number line. These roots divide the number line into intervals.
Choose a test value within each interval and substitute it back into the original inequality.
Determine which intervals satisfy the inequality. The solution set includes these intervals. Pay attention to whether the inequality is strict () or non-strict (=). For non-strict inequalities, the roots themselves are part of the solution.
Quadratic Formula: For ax^2 + bx + c = 0, the roots are x = [-b ± sqrt(b^2 - 4ac)] / 2a.
Variables Table
Variable
Meaning
Unit
Typical Range
x
The unknown variable being solved for.
Depends on context (e.g., units, abstract number)
(-∞, ∞) for real numbers
a, b, c
Coefficients and constants in the inequality.
Depends on context
Real numbers
Roots
Values of x where the expression equals zero.
Same as x
Real or complex numbers
Intervals
Segments of the number line defined by roots.
N/A
e.g., (-∞, r1), (r1, r2), (r2, ∞)
How to Use This Inequality Calculator Step by Step
Using this inequality calculator step by step is straightforward:
Input the Inequality: In the "Enter Inequality" field, type your inequality precisely. Use 'x' for the variable. For quadratic terms, use '^' (e.g., x^2). Ensure you use the correct inequality symbols: <, >, <=, >=, or =.
Select Variable Type: Choose whether you need solutions within the set of all real numbers or only integers.
Click Calculate: The calculator will process your input.
Review Results:
Primary Result: This shows the main solution, typically in interval notation (e.g., (-∞, 5)).
Intermediate Values: You'll see the solution in set notation (e.g., {x | x ∈ ℝ, x < 5}) and the count of integer solutions if applicable.
Calculation Steps: A table details each algebraic manipulation performed to reach the solution.
Chart: A visual graph displays the number line and highlights the solution set.
Understand the Steps: Use the table to follow the logic and learn how each step contributes to isolating the variable or defining the solution intervals.
Decision Making: The results help you determine the range of values for 'x' that satisfy the condition. For example, if solving CostPerItem * x + FixedCost > Revenue, the solution tells you the minimum number of items 'x' you need to sell to make a profit.
Reset: Click "Reset" to clear all fields and start over with a new inequality.
Copy Results: Use "Copy Results" to save the primary result, intermediate values, and key assumptions for later use.
Practical Examples (Real-World Use Cases)
Example 1: Linear Inequality – Budgeting
Scenario: Sarah has a budget of $500 for a party. She needs to rent a DJ for $150 and wants to buy snacks for $5 per person. How many people can she invite?
Inequality:5p + 150 <= 500 (where 'p' is the number of people)
Inputs for Calculator:
Inequality: 5p + 150 <= 500
Variable Type: Integer (since you can't invite a fraction of a person)
Calculator Output (Simulated):
Primary Result: p <= 70
Intermediate Values:
Interval Notation: (-∞, 70]
Set Notation: {p | p ∈ ℤ, p ≤ 70}
Number of Integer Solutions: Infinite (but practically limited by context, e.g., non-negative)
Key Assumption: The variable 'p' represents the number of people and must be a non-negative integer. The calculator provides the mathematical upper bound.
Interpretation: Sarah can invite up to 70 people to stay within her $500 budget.
Example 2: Quadratic Inequality – Profit Maximization
Scenario: A company's weekly profit P (in thousands of dollars) from selling x units of a product is given by P(x) = -x^2 + 12x - 20. For what number of units sold will the company make a profit of at least $16,000?
Inequality:-x^2 + 12x - 20 >= 16 (since P is in thousands)
Inputs for Calculator:
Inequality: -x^2 + 12x - 20 >= 16
Variable Type: Real Numbers (initially, then consider practical integer units)
Calculator Output (Simulated):
First, rewrite as -x^2 + 12x - 36 >= 0.
Roots of -x^2 + 12x - 36 = 0 are found using the quadratic formula or factoring: -(x^2 - 12x + 36) = 0 => -(x - 6)^2 = 0 => x = 6 (a repeated root).
Primary Result: x = 6
Intermediate Values:
Interval Notation: [6, 6] (or just {6})
Set Notation: {x | x ∈ ℝ, x = 6}
Number of Integer Solutions: 1
Key Assumption: The profit function is a downward-opening parabola. The inequality asks for profit >= 16 (thousand).
Interpretation: The company achieves exactly $16,000 profit only when selling exactly 6 units. To make *at least* $16,000, they must sell precisely 6 units. If the inequality was slightly different, yielding a range, we'd interpret that range.
Key Factors That Affect Inequality Results
While the calculator automates the process, understanding the factors influencing the results is key for accurate application:
Type of Inequality: Linear inequalities are generally simpler, solved by isolating the variable. Quadratic and higher-order polynomial inequalities require finding roots and testing intervals, leading to potentially complex solution sets (intervals, unions of intervals).
Inequality Symbol: The symbol (<, >, <=, >=) dictates whether the boundary points (roots or isolated variable values) are included in the solution set. Strict inequalities exclude boundaries, while non-strict ones include them.
Coefficients and Constants: The specific numerical values (a, b, c) determine the location of roots and the shape/position of the graph (for polynomial inequalities), directly impacting the solution intervals.
Variable Type (Real vs. Integer): The calculator's setting significantly changes the output. A real number solution might be an interval like (2, 5), while an integer solution would be {3, 4}. This is critical in practical applications where only whole units or discrete values make sense.
Domain Restrictions: Sometimes, the context of a problem imposes restrictions not explicitly stated in the inequality itself. For example, the number of items sold cannot be negative. The calculator provides the mathematical solution, but you must apply real-world constraints.
Graphing Accuracy: Visualizing the inequality on a number line or coordinate plane (as the chart does) helps confirm the algebraic solution. Understanding the relationship between the function's graph and the inequality sign is crucial. For f(x) > 0, you look for where the graph is above the x-axis.
Algebraic Errors: Mistakes in simplification, especially when multiplying/dividing by negatives or handling square roots, can lead to incorrect solutions. The step-by-step breakdown helps catch these.
Frequently Asked Questions (FAQ)
What's the difference between solving an equation and an inequality?
Solving an equation (e.g., 2x + 1 = 5) typically yields a single value or a finite set of values for the variable. Solving an inequality (e.g., 2x + 1 < 5) usually results in a range of values (an interval or a set of intervals) that satisfy the condition.
When do I flip the inequality sign?
You must reverse the direction of the inequality sign (e.g., < becomes >) whenever you multiply or divide both sides of the inequality by a negative number.
How do I handle inequalities with fractions?
For inequalities with fractions (rational inequalities), you typically need to find a common denominator or multiply by the square of the denominator (to ensure positivity) to clear the fractions. Then, proceed as with polynomial inequalities, being careful about values that make the original denominator zero.
What does interval notation mean?
Interval notation is a way to represent a range of numbers. Parentheses ( ) indicate that the endpoint is not included (for strict inequalities or infinity), while square brackets [ ] indicate that the endpoint is included (for non-strict inequalities). For example, (2, 5] means all numbers greater than 2 and less than or equal to 5.
Can this calculator solve absolute value inequalities?
This specific calculator is designed for linear and standard quadratic inequalities. Absolute value inequalities (e.g., |x - 3| < 5) require a different approach, often splitting into two separate inequalities. You might need a specialized calculator for those.
What if the quadratic inequality has no real roots?
If the corresponding quadratic equation ax^2 + bx + c = 0 has no real roots (the discriminant b^2 - 4ac is negative), the quadratic expression ax^2 + bx + c will always have the same sign (either always positive or always negative, determined by the sign of 'a'). The inequality will either be true for all real numbers or for no real numbers.
How does the 'Integer' option work?
When 'Integer' is selected, the calculator finds the mathematical solution set (usually intervals) and then filters it to include only whole numbers (positive, negative, and zero) that fall within that set. It also provides a count of these integer solutions.
Can I use variables other than 'x'?
Currently, this calculator is configured to work specifically with the variable 'x'. You would need to substitute your variable for 'x' when entering the inequality.
What does the chart represent?
The chart typically displays a number line. For linear inequalities, it highlights the segment representing the solution. For quadratic inequalities, it might show the parabola and shade the regions above or below the x-axis corresponding to the inequality, or simply highlight the solution intervals on the number line.