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Inequality Solver Calculator

Solve linear inequalities with one variable and find the solution set. This calculator helps you solve inequalities of the form ax + b < c, ax + b ≤ c, ax + b> c, or ax + b ≥ c.

Enter the coefficient of x in your inequality
Enter the constant term on the left side of the inequality
Enter the constant on the right side of the inequality

How to Use This Calculator

  1. Enter the coefficient of x (a) in your inequality
  2. Enter the constant term (b) on the left side of the inequality
  3. Select the appropriate inequality symbol (<, ≤,>, ≥)
  4. Enter the constant term (c) on the right side of the inequality
  5. Click "Solve" to find the solution set

Formula Used

ax + b < c → x < (c - b)/a
ax + b ≤ c → x ≤ (c - b)/a
ax + b > c → x > (c - b)/a
ax + b ≥ c → x ≥ (c - b)/a

Where:

  • a = coefficient of x
  • b = constant term on the left side
  • c = constant term on the right side

Note: When dividing by a negative number, the inequality sign reverses.

Example Calculation

Real-World Scenario:

A company produces t-shirts at a cost of $5 per shirt plus a fixed cost of $100. They want to determine how many shirts they need to sell to make a profit of at least $500 if each shirt sells for $15.

Given:

  • Revenue = 15x (where x is the number of shirts)
  • Cost = 5x + 100
  • Profit = Revenue - Cost = 15x - (5x + 100) = 10x - 100

Inequality:

10x - 100 ≥ 500

Calculation:

10x ≥ 600

x ≥ 60

Result: The company needs to sell at least 60 shirts to make a profit of $500 or more.

Why This Calculation Matters

Practical Applications

  • Determining break-even points in business
  • Finding minimum or maximum values in optimization problems
  • Setting constraints in engineering design
  • Calculating dosage ranges in medicine

Key Benefits

  • Quickly solve linear inequalities without manual calculations
  • Visualize solution sets on a number line
  • Understand the relationship between variables in constraints
  • Apply mathematical concepts to real-world problems

Common Mistakes & Tips

When dividing or multiplying both sides of an inequality by a negative number, you must reverse the inequality sign. For example, if you have -2x > 4, dividing by -2 gives x < -2 (notice the sign reversal).

Remember that x > 3 means all numbers greater than 3 (not including 3), while x ≥ 3 includes 3 and all numbers greater than 3. Similarly, x < 3 means all numbers less than 3 (not including 3), while x ≤ 3 includes 3 and all numbers less than 3.

Be aware of special cases like when a = 0. For example, if you have 0x + 3 > 5, this simplifies to 3 > 5, which is false, so there is no solution. If you have 0x + 3 < 5, this simplifies to 3 < 5, which is true, so all real numbers are solutions.

Frequently Asked Questions

An equation uses an equals sign (=) and has a specific solution or set of solutions. An inequality uses symbols like <, ≤,>, or ≥ and typically has a range of solutions. For example, x = 3 has only one solution (3), while x > 3 has infinitely many solutions (all numbers greater than 3).

Reversing the inequality sign when dividing by a negative number maintains the truth of the statement. For example, if 6 > 2, dividing both sides by -2 gives -3 < -1 (notice the sign reversal). Without reversing the sign, we would incorrectly conclude that -3> -1, which is false.

To graph the solution to a linear inequality on a number line: 1) Find the boundary point by solving the corresponding equation, 2) Draw a point on the number line at this boundary value, 3) Use an open circle for < or> and a closed circle for ≤ or ≥, 4) Shade the number line in the direction of the inequality (to the left for < or ≤, to the right for> or ≥).

References & Disclaimer

Mathematical Disclaimer

This calculator provides solutions to linear inequalities based on standard mathematical principles. While we strive for accuracy, users should verify critical calculations independently. This tool is for educational purposes and should not be used as a substitute for professional mathematical consultation.

References

Accuracy Notice

This calculator solves linear inequalities of the form ax + b < c, ax + b ≤ c, ax + b> c, or ax + b ≥ c. It does not handle more complex inequalities involving absolute values, quadratic terms, or rational expressions. Results are rounded to 4 decimal places for display purposes.

About the Author

Kumaravel Madhavan

Web developer and data researcher creating accurate, easy-to-use calculators across health, finance, education, and construction and more. Works with subject-matter experts to ensure formulas meet trusted standards like WHO, NIH, and ISO.

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