Inequality Calculator

What Is an Inequality?

An inequality is a mathematical expression that relates two values using symbols such as <, ≤, >, and ≥. Inequalities express that one value is either less than, less than or equal to, greater than, or greater than or equal to another value. For example:

• x < 5: x is less than 5
• 3 ≤ y: y is greater than or equal to 3
• z > 7: z is greater than 7

Inequalities are fundamental in algebra and are used to represent ranges of possible values for variables.

Types of Inequalities

• Linear Inequalities: A linear inequality involves variables raised only to the first power (i.e., ax + b < c).
• Non-linear Inequalities: These involve variables raised to powers other than one, such as quadratic, cubic, or rational inequalities.
• Compound Inequalities: A compound inequality consists of two or more inequalities combined by “and” or “or.”
• Absolute Value Inequalities: Inequalities that include absolute value expressions.

Formal Definition of Inequality

The general form of a linear inequality is:

ax + b < c or ax + b ≥ c

To solve an inequality, you typically isolate the variable by performing the same operations on both sides of the inequality, similar to solving equations.

Why It Matters

• 🔓 Problem Solving: Inequalities help represent and solve real-world problems, such as finding acceptable ranges of values or defining constraints in optimization.
• 📉 Graphing: Inequalities form the basis for shading regions on graphs, which helps visualize solution sets.
• 🔬 Applications in Science & Engineering: Inequalities are used to describe limits and bounds in fields like physics, economics, and engineering.

How to Use the Inequality Calculator

1. Enter the Inequality: Type the inequality you want to solve. For example, 2x + 3 > 7 or x² - 5x + 6 ≤ 0.
2. Select the Type: Choose the type of inequality (linear, non-linear, absolute value, or compound).
3. Click "Solve": The calculator will process your input and give you the solution.
4. View Step-by-Step Solution: The solution will include each step taken to solve the inequality, so you can learn the process.

Inequality Examples

• Example 1: Solve 2x - 3 < 7

  Solution: Add 3 to both sides: 2x < 10. Divide both sides by 2: x < 5

• Example 2: Solve x² - 4x + 3 ≥ 0

  Solution: Factor the quadratic: (x - 1)(x - 3) ≥ 0. The solution is x ≤ 1 or x ≥ 3

• Example 3: Solve |x + 2| < 4

  Solution: Split into two inequalities: x + 2 < 4 and x + 2 > -4. Solving these gives -6 < x < 2

• Example 4: Solve the compound inequality -2 ≤ x + 1 < 3

  Solution: Subtract 1 from all parts: -3 ≤ x < 2

Applications of Inequalities

• 🔢 Optimization Problems: Inequalities define the limits of possible solutions in optimization.
• ⚖️ Economics: Used to model supply and demand, profit margins, and more.
• 🔒 Security & Encryption: Inequalities are used in cryptography to define secure key spaces.
• 🏋️ Engineering: Represent tolerances, stress limits, and safety factors.

Frequently Asked Questions (FAQ)