Cal11 calculator
Search calculator...

Solve Inequalities Interval Notation Calculator

Reviewed by Calculator Editorial Team

This calculator helps you solve linear and quadratic inequalities and displays the solutions in interval notation. Whether you're a student learning algebra or a professional needing quick reference, this tool provides clear, step-by-step solutions.

How to Use This Calculator

Using the inequality solver is straightforward:

  1. Enter your inequality in the input field. For example, type x + 3 > 5 or x² - 4x < 4.
  2. Select the type of inequality (linear or quadratic).
  3. Click "Calculate" to see the solution in interval notation.
  4. Review the detailed steps and graph (if available) to understand how the solution was derived.

The calculator will display the solution set in interval notation, such as (-∞, 2) or (-2, 3), and provide a graphical representation when possible.

How to Solve Inequalities

Solving inequalities involves similar steps to solving equations, but with additional considerations for the inequality sign and the concept of direction.

Step 1: Isolate the Variable

Move all terms containing the variable to one side and constants to the other. For example, solve 3x - 5 > 10:

  1. Add 5 to both sides: 3x > 15.
  2. Divide both sides by 3: x > 5.

Step 2: Consider the Inequality Sign

When multiplying or dividing both sides by a negative number, reverse the inequality sign. For example, solve -2x + 4 < 10:

  1. Subtract 4 from both sides: -2x < 6.
  2. Divide both sides by -2 and reverse the sign: x > -3.

Step 3: Express the Solution in Interval Notation

Convert the inequality to interval notation. For x > 5, the solution is (5, ∞). For x < -3, it's (-∞, -3).

Understanding Interval Notation

Interval notation is a concise way to represent sets of real numbers. The key symbols are:

  • (a, b) - All numbers between a and b, not including a and b.
  • [a, b] - All numbers between a and b, including a and b.
  • (a, ∞) - All numbers greater than a.
  • (-∞, b) - All numbers less than b.
  • (-∞, ∞) - All real numbers.

For example, the solution to x² < 9 is (-3, 3), meaning all numbers between -3 and 3, not including -3 and 3.

Worked Examples

Example 1: Linear Inequality

Solve 2x + 3 > 7:

  1. Subtract 3 from both sides: 2x > 4.
  2. Divide both sides by 2: x > 2.
  3. Solution in interval notation: (2, ∞).

Example 2: Quadratic Inequality

Solve x² - 5x + 6 < 0:

  1. Factor the quadratic: (x - 2)(x - 3) < 0.
  2. Find critical points: x = 2 and x = 3.
  3. Test intervals: The inequality holds for 2 < x < 3.
  4. Solution in interval notation: (2, 3).

Frequently Asked Questions

What types of inequalities can this calculator solve?

This calculator can solve linear inequalities (e.g., ax + b > c) and quadratic inequalities (e.g., ax² + bx + c < 0).

How do I interpret the interval notation solution?

Interval notation shows the range of values that satisfy the inequality. For example, (-2, 5) means all numbers greater than -2 and less than 5.

What if the inequality has no solution?

If the inequality is always false (e.g., x > x), the calculator will indicate that there is no solution.

Can I solve compound inequalities with this calculator?

Yes, you can enter compound inequalities like 1 < x < 5, and the calculator will solve them step-by-step.