Solving Inequalities Interval Notation Calculator

This calculator helps you solve inequalities and express the solutions in interval notation. Learn how to convert mathematical inequalities to interval notation and visualize the solution sets.

How to Use This Calculator

Enter your inequality in the input field, select the type of inequality, and click "Calculate". The calculator will solve the inequality and display the solution in interval notation.

The calculator supports basic linear inequalities. For more complex inequalities, you may need to solve them manually or use a more advanced mathematical tool.

What Is Interval Notation?

Interval notation is a way to represent a set of real numbers that lie between two endpoints. It is commonly used in mathematics to describe the solution set of an inequality.

There are several types of interval notation:

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

Solving Inequalities

To solve an inequality, follow these steps:

Isolate the variable on one side of the inequality.
Perform the same operation on both sides to maintain the inequality.
Determine the solution set based on the inequality sign.
Express the solution set in interval notation.

Remember that when you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign.

Converting to Interval Notation

Once you have solved the inequality, you can convert the solution set to interval notation. Here are some examples:

If the solution is x > 3, the interval notation is (3, ∞).
If the solution is x ≤ 5, the interval notation is (-∞, 5].
If the solution is 2 < x < 8, the interval notation is (2, 8).
If the solution is x ≥ -4, the interval notation is [-4, ∞).

Examples

Example 1: Solving 2x + 5 > 11

Step 1: Subtract 5 from both sides: 2x > 6

Step 2: Divide both sides by 2: x > 3

Solution in interval notation: (3, ∞)

Example 2: Solving -4x ≤ 20

Step 1: Divide both sides by -4 (remember to reverse the inequality sign): x ≥ -5

Solution in interval notation: [-5, ∞)

Example 3: Solving -3 < 2x + 4 < 7

Step 1: Subtract 4 from all parts: -7 < 2x < 3

Step 2: Divide all parts by 2: -3.5 < x < 1.5

Solution in interval notation: (-3.5, 1.5)

FAQ

What types of inequalities can this calculator solve?

This calculator can solve basic linear inequalities. For more complex inequalities, you may need to solve them manually or use a more advanced mathematical tool.

How do I interpret the interval notation?

Interval notation represents a set of real numbers between two endpoints. The parentheses ( ) indicate that the endpoint is not included, while the brackets [ ] indicate that the endpoint is included.

What if I get a different solution than expected?

Double-check your inequality and the steps you took to solve it. Remember that multiplying or dividing by a negative number requires reversing the inequality sign. If you're still unsure, consult a math textbook or ask a teacher for help.