Solve Inequalities with One Variable

One-Variable Inequality Solver

Enter a linear inequality (example: 2x − 5 > 7) and click Solve to see the solution steps.

Solution will appear here.

Note: This tool demonstrates the solving process. Always pay attention to inequality signs, especially when multiplying or dividing by negatives.

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Solve Inequalities – Complete Explanation

An inequality is a mathematical statement that compares two expressions using symbols such as <, >, ≤, or ≥. Solving inequalities with one variable means finding all values of the variable that make the inequality true.

This comprehensive guide will take you from basic inequality concepts through advanced problem-solving techniques, with plenty of examples and practice opportunities.

1. What Is an Inequality?

Unlike equations, inequalities do not have a single solution. Instead, they usually have a range of solutions.

Examples:

x > 3
2x − 5 ≤ 7
−4x < 12

2. Inequality Symbols

• > greater than
• < less than
• ≥ greater than or equal to
• ≤ less than or equal to

3. Core Rule for Solving Inequalities

Solving inequalities follows the same steps as solving equations, with one critical difference:

Important Rule: When you multiply or divide both sides by a negative number, you must reverse the inequality sign.

For example:

-2x < 6
x > -3  (divided by -2, sign reversed)

4. Step-by-Step Examples

Example 1: Simple Inequality

x + 4 > 9

Subtract 4 from both sides:

x > 5

Solution: All numbers greater than 5.

Example 2: With Multiplication

3x ≤ 12

Divide both sides by 3:

x ≤ 4

Solution: All numbers less than or equal to 4.

Example 3: With Negative Coefficient

−2x > 6

Divide both sides by −2 and reverse the sign:

x < −3

Solution: All numbers less than -3.

Example 4: Two-Step Inequality

2x − 5 > 7

Add 5 to both sides:

2x > 12

Divide by 2:

x > 6

Solution: All numbers greater than 6.

5. Solving Inequalities with Variables on Both Sides

4x − 3 > 2x + 5

Move variable terms to one side:

4x − 2x − 3 > 5
2x − 3 > 5

Add 3 to both sides:

2x > 8

Divide by 2:

x > 4

6. Inequalities with Fractions

(x/2) + 3 < 5

Subtract 3 from both sides:

x/2 < 2

Multiply both sides by 2:

x < 4

7. Graphing Solutions on a Number Line

Inequality solutions are often shown on a number line:

• Open circle (○) → < or > (value not included)
• Closed circle (●) → ≤ or ≥ (value included)
• Arrow shows direction of solutions

Examples:

x > 3  →  open circle at 3, arrow to the right
x ≤ 2  →  closed circle at 2, arrow to the left

8. Compound Inequalities

Compound inequalities combine two inequalities with "and" or "or".

Example with "And"

x > 2 and x < 6

This can be written as:

2 < x < 6

Solution: Numbers between 2 and 6 (exclusive).

Example with "Or"

x < 1 or x > 4

Solution: Numbers less than 1 OR greater than 4.

9. Special Cases

No Solution

x > 5 and x < 3

No number satisfies both conditions.

All Real Numbers

x > 2 or x < 5

Every real number satisfies at least one condition.

10. Common Mistakes to Avoid

• Forgetting to reverse the inequality sign when multiplying/dividing by a negative
• Mixing up < and ≤ (including/excluding the endpoint)
• Stopping before isolating the variable completely
• Incorrect arithmetic when moving terms
• Graphing errors (wrong direction, wrong circle type)
• Misinterpreting compound inequalities

11. Checking Your Solution

Always verify your answer by substituting a value from your solution set:

Inequality: 2x - 5 > 7
Solution: x > 6

Test x = 7:
2(7) - 5 = 14 - 5 = 9 > 7 ✓

Test x = 5 (should NOT work):
2(5) - 5 = 10 - 5 = 5 > 7? No ✓

12. Practice Problems with Solutions

Problem 1: Solve x − 7 ≥ 2

Solution: x ≥ 9

Check: x = 9: 9-7=2 ✓; x = 10: 10-7=3≥2 ✓

Problem 2: Solve 5x < 20

Solution: x < 4

Check: x = 3: 15<20 ✓; x = 4: 20<20? No ✓

Problem 3: Solve −3x ≤ 9

Solution: x ≥ −3 (sign reversed!)

Check: x = -2: -3(-2)=6≤9 ✓; x = -4: -3(-4)=12≤9? No ✓

Problem 4: Solve 2x + 3 > 4x - 5

Solution: 2x - 4x > -5 - 3 → -2x > -8 → x < 4

Problem 5: Solve -4 ≤ 2x < 8

Solution: Divide all parts by 2: -2 ≤ x < 4

13. Real-World Applications

One-variable inequalities are used extensively in real life:

• Budget limits: Spending ≤ available money
• Speed restrictions: Speed ≤ speed limit
• Age requirements: Age ≥ minimum age
• Weight limits: Weight ≤ maximum capacity
• Temperature ranges: Min temp ≤ actual ≤ max temp
• Profit analysis: Revenue > costs for profit
• Engineering safety: Stress ≤ material strength
• Health metrics: BMI in healthy range

Example: Budget Planning

If you have $100 to spend and each item costs $15, how many items can you buy?

15x ≤ 100
x ≤ 6.67
x ≤ 6 (since you can't buy a fraction)

14. Advanced Topics

Inequalities with Absolute Value

|x| < 5  →  -5 < x < 5
|x| > 5  →  x < -5 or x > 5

Quadratic Inequalities

x² - 4 > 0
(x - 2)(x + 2) > 0
x < -2 or x > 2

15. Tips for Mastering Inequalities

• Always identify the inequality symbol first
• Treat it like an equation until the final step
• Remember the sign reversal rule for negatives
• Graph your solution to visualize the range
• Test boundary values and points in each region
• Practice with all four inequality types
• Use real-world problems to build intuition

16. Final Thoughts

Solving inequalities with one variable is a foundational algebra skill. It teaches logical reasoning, careful attention to rules, and prepares you for systems of inequalities, linear programming, and more advanced mathematics.

The key to mastery is practice with many variations and always checking your work. Use the calculator above to verify your solutions, but make sure you understand each step of the process.

With consistent practice, solving inequalities will become second nature, and you'll be ready to tackle more complex mathematical challenges.