Solve System of Inequalities
System of Inequalities Solver
Enter multiple inequalities (one per line), for example:
x > 2 x ≤ 5
System of Inequalities – Complete Explanation
A system of inequalities is a group of two or more inequalities that are solved together. The solution is the set of all values that satisfy every inequality in the system.
This comprehensive guide will take you through everything you need to know about solving systems of inequalities, from basic concepts to advanced applications.
1. What Is a System of Inequalities?
Just like systems of equations, systems of inequalities involve multiple conditions. However, instead of a single value, the solution is usually a range of values (or region) that satisfies all conditions simultaneously.
Example:
x > 1 x < 6
The solution is:
1 < x < 6
This means x can be any number between 1 and 6 (but not including 1 and 6).
2. Solving Systems with One Variable
When a system has one variable, the steps are straightforward:
- Solve each inequality individually using standard inequality rules
- Write each solution as an interval using interval notation
- Find the intersection (overlap) of all intervals
- The overlapping region is the final solution
Interval Notation Review
- (a, b) → all numbers between a and b, exclusive (a < x < b)
- [a, b] → all numbers between a and b, inclusive (a ≤ x ≤ b)
- (a, ∞) → all numbers greater than a (x > a)
- (-∞, b] → all numbers less than or equal to b (x ≤ b)
3. Step-by-Step Examples
Example 1: Simple System
x ≥ 2 x < 7
Step 1: Solve each inequality
x ≥ 2 → [2, ∞) x < 7 → (-∞, 7)
Step 2: Find the overlap
Overlap: [2, ∞) ∩ (-∞, 7) = [2, 7)
Step 3: Write in inequality form
2 ≤ x < 7
This means x can be any number from 2 up to (but not including) 7.
Example 2: With Negative Numbers
x > −4 x ≤ 1
Step 1: Solve each inequality
x > -4 → (-4, ∞) x ≤ 1 → (-∞, 1]
Step 2: Find the overlap
(-4, ∞) ∩ (-∞, 1] = (-4, 1]
Step 3: Write in inequality form
-4 < x ≤ 1
Example 3: Three Inequalities
x ≥ 0 x > 2 x ≤ 8
Step 1: Solve each inequality
x ≥ 0 → [0, ∞) x > 2 → (2, ∞) x ≤ 8 → (-∞, 8]
Step 2: Find the overlap of all three
[0, ∞) ∩ (2, ∞) ∩ (-∞, 8] = (2, 8]
Step 3: Write in inequality form
2 < x ≤ 8
Notice that x ≥ 0 was automatically satisfied since x > 2 is stricter.
Example 4: No Solution
x > 5 x < 2
Step 1: Solve each inequality
x > 5 → (5, ∞) x < 2 → (-∞, 2)
Step 2: Find the overlap
(5, ∞) ∩ (-∞, 2) = ∅ (empty set)
Result: No solution
There is no number that is simultaneously greater than 5 and less than 2.
Example 5: All Real Numbers
x > -10 x < 100
Step 1: Solve each inequality
x > -10 → (-10, ∞) x < 100 → (-∞, 100)
Step 2: Find the overlap
(-10, ∞) ∩ (-∞, 100) = (-10, 100)
Step 3: Write in inequality form
-10 < x < 100
4. Graphing a System of Inequalities on a Number Line
Visualizing the solution helps build intuition:
- Graph each inequality separately on the same number line
- Use open circles (○) for < or > (endpoint not included)
- Use closed circles (●) for ≤ or ≥ (endpoint included)
- The overlapping shaded region is the solution
Example: 2 < x ≤ 5
Number line representation:
○===============●
----|-----|-----|-----|----
2 3 4 5
Open circle at 2, closed circle at 5, shaded in between.
5. Systems with Two Variables (Introduction)
Systems of inequalities can also involve two variables:
y > x + 1 y ≤ −x + 5
In this case, solutions are shown on a coordinate plane:
- Graph each inequality as a boundary line (dashed for < or >, solid for ≤ or ≥)
- Shade the region that satisfies each inequality
- The overlapping shaded region is the solution set
- Any point in the overlapping region satisfies all inequalities
Example: Two-Variable System
y ≥ x y < 3
The solution is all points where y is greater than or equal to x AND y is less than 3.
6. Special Cases
Case 1: One Inequality is Always True
x > -1000 x < 5
Since x > -1000 is almost always true (except for very small numbers), the solution is essentially x < 5.
Case 2: One Inequality is Always False
x > 10 x < 5
No overlap → No solution.
Case 3: Identical Inequalities
x ≥ 3 x ≥ 3
The solution is simply x ≥ 3.
7. Common Mistakes to Avoid
- Forgetting to solve each inequality separately before combining
- Missing the overlap region when intervals don't align perfectly
- Incorrect inequality signs when writing the final solution
- Mixing open and closed intervals (using wrong circle type)
- Assuming "and" always means intersection (it does, but "or" means union)
- Not checking boundary values when testing your solution
- Graphing errors when visualizing on a number line
8. Practice Problems with Solutions
Problem 1: Solve x > 0 and x ≤ 4
Solution: 0 < x ≤ 4
Check: x = 2 works (2>0 and 2≤4); x = 0 doesn't work (not >0); x = 4 works (4≤4)
Problem 2: Solve x ≥ −2 and x ≥ 3
Solution: x ≥ 3 (since ≥3 is stricter than ≥-2)
Check: x = 5 works (≥ both); x = 0 doesn't work (not ≥3)
Problem 3: Solve x < −1 and x > 5
Solution: No solution
Check: No number can be both <-1 and >5
Problem 4: Solve x ≤ 10 and x ≥ 2 and x < 7
Solution: 2 ≤ x < 7
Check: x = 5 works; x = 1 doesn't work (not ≥2); x = 8 doesn't work (not <7)
Problem 5: Solve x > -5 and x < 10 and x ≥ 0
Solution: 0 ≤ x < 10
Check: x = 0 works (≥0); x = -3 doesn't work (not ≥0); x = 15 doesn't work (not <10)
9. Real-World Applications
Systems of inequalities are used extensively in real-world scenarios:
- Budget constraints: Total cost ≤ budget AND each item cost ≥ 0
- Scheduling and planning: Time constraints with multiple conditions
- Engineering safety limits: Temperature, pressure, and stress must all be within safe ranges
- Optimization problems: Finding the best solution given multiple constraints
- Economics and production models: Resource limitations and production targets
- Health guidelines: Age, weight, and health metrics within recommended ranges
- Quality control: Multiple measurements must fall within specification limits
Example: Manufacturing Constraints
A factory produces items that must meet these conditions:
Weight ≥ 50 grams Weight ≤ 200 grams Length ≥ 10 cm Length ≤ 30 cm
The solution is the range of acceptable items that satisfy all four inequalities simultaneously.
Example: Event Planning
An event requires:
Attendees ≥ 50 (minimum to break even) Attendees ≤ 200 (venue capacity) Budget per person ≤ $50
The number of attendees must satisfy all three conditions.
10. Tips for Mastering Systems of Inequalities
- Always solve each inequality separately first
- Write solutions in interval notation for clarity
- Draw a number line to visualize the overlap
- Test boundary values to ensure correct inequality signs
- Practice with 2, 3, and 4 inequalities to build skill
- Remember that "and" means intersection (both true)
- For "or" problems, you'd take the union instead
- Check your answer by picking a value from the solution set
- Also check values just outside to confirm they don't work
11. Connection to Linear Programming
Systems of inequalities form the foundation of linear programming, where we find the optimal solution (maximum or minimum) subject to multiple constraints.
For example:
Maximize profit = 5x + 3y Subject to: x ≥ 0 y ≥ 0 x + y ≤ 100 2x + y ≤ 150
The feasible region (where all inequalities are satisfied) contains all possible solutions, and the optimal solution is found at a corner point of this region.
12. Final Thoughts
Solving systems of inequalities teaches you how to handle multiple conditions at once. This skill is essential for algebra, graphing, optimization, and real-world decision making.
The key to mastery is practice with many different combinations—some with clear overlap, some with no solution, and some where one inequality dominates the others.
Use the solver above to visualize the process and check your work, but always practice finding the overlapping solution manually to build strong mathematical understanding. Once you master systems of inequalities, you'll be ready for linear programming, multivariable optimization, and advanced applied mathematics.