Inequalities Solver & Grapher - Solve Inequalities
Solve linear inequalities and view solutions in interval notation. Get step-by-step solutions with number line representations.
Linear Inequality
Solution Steps
Number Line Representation
Solution
What is an Inequalities Solver?
An Inequalities Solver is a mathematical tool that finds the range of values that satisfy an inequality statement. Unlike equations that have specific solutions, inequalities have solution sets represented as intervals.
This calculator solves:
- Linear Inequalities - In the form ax + b < c (or >, ≤, ≥)
- Interval Notation - Displays solutions as mathematical intervals
- Number Line Graphs - Visual representation of solution sets
To solve systems of equations, try our Simultaneous Equations Solver with step-by-step solutions.
For quadratic equations, check our Quadratic Equation Solver with discriminant analysis.
To work with linear equations, use our Slope & Intercept Calculator for line equations.
How to Solve Linear Inequalities
For a linear inequality ax + b < c, follow these steps:
Step 2: Divide by a to isolate x
Step 3: If a < 0, flip the inequality sign
Example: 2x + 3 < 11
- Subtract 3: 2x < 8
- Divide by 2: x < 4
- Interval notation: (-∞, 4)
Understanding Interval Notation
Interval notation uses brackets and parentheses to represent solution sets:
Parentheses ( )
Endpoint NOT included
x < 4 → (-∞, 4)
Brackets [ ]
Endpoint IS included
x ≤ 4 → (-∞, 4]
Infinity Symbol
Always use parentheses
(-∞, 4) or [2, ∞)
Compound
Between two values
2 ≤ x < 5 → [2, 5)
How to Use This Calculator
Enter Coefficient
Input the value of 'a' in ax + b
Enter Constant
Input the value of 'b' in ax + b
Choose Comparator
Select <, ≤, >, or ≥
Enter Value
Input 'c' and click Solve
Benefits of Using This Calculator
- • Step-by-Step Solutions: See every step in solving the inequality.
- • Interval Notation: Get proper mathematical notation for solutions.
- • Number Line Graphs: Visual representation helps understand solution sets.
- • Sign Flip Detection: Automatically handles negative coefficient division.
When to Flip the Inequality Sign
1. Dividing by Negative
When you divide or multiply both sides by a negative number, flip the sign.
-2x < 6 → x > -3
2. Multiplying by Negative
Same rule applies when multiplying by negative values.
-x/3 > 2 → -x > 6 → x < -6
3. Keep Sign Otherwise
Adding, subtracting, or dividing by positive numbers keeps the sign.
2x + 5 < 11 → 2x < 6 → x < 3
Frequently Asked Questions (FAQ)
Q: What is an inequality in mathematics?
A: An inequality is a mathematical statement that shows the relationship between two expressions that are not equal, using symbols like < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to).
Q: When do you flip the inequality sign?
A: You must flip the inequality sign when multiplying or dividing both sides by a negative number. For example, if -2x < 6, dividing by -2 gives x > -3 (the sign flips from < to >).
Q: What is interval notation?
A: Interval notation is a way to represent solution sets of inequalities. For example, x < 4 is written as (-∞, 4), x ≥ 2 is written as [2, ∞), and 2 ≤ x < 5 is written as [2, 5). Parentheses ( ) mean not included, brackets [ ] mean included.
Q: How do you solve a linear inequality?
A: To solve a linear inequality like ax + b < c: isolate the variable by subtracting b from both sides, then divide by a. Remember to flip the inequality sign if you divide by a negative number.
Q: What is the difference between < and ≤?
A: < means "less than" and does not include the boundary value (open interval), while ≤ means "less than or equal to" and includes the boundary value (closed interval). For example, x < 5 means all values up to but not including 5, while x ≤ 5 includes 5.