Inequality Calculator

Understanding Inequalities

What is a Inequality?

A inequality is a mathematical statement that relates a expression as either less than or greater than another. Unlike equations, which show equality, inequalities show a relationship of imbalance. Inequalities are similar to equations but use inequality signs (>, <, ≥, ≤) instead of an equals sign.

Forms of Inequalities

Inequalities can be written in several forms, including:

• Standard form: ax + b > c
• Slope-intercept form: y > mx + b
• General form: Ax + By + C > 0

Our calculator focuses on the standard form, which is one of the most common ways to express inequalities.

Solving Inequalities

Solving inequalities follows a similar process to solving equations, with one critical exception: when you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign.

Step-by-step process:

1. Isolate the variable term on one side of the inequality
2. Combine like terms if necessary
3. Divide both sides by the coefficient of the variable
4. Remember to reverse the inequality sign if dividing by a negative number
5. Express the solution in inequality notation, interval notation, or on a number line

Example Problems and Solutions

Example 1: Solve 2x + 3 > 7

Step 1: Subtract 3 from both sides: 2x > 4

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

Thus, the solution is x > 2, which means all real numbers greater than 2.

Example 2: Solve -3x + 5 ≤ 11

Step 1: Subtract 5 from both sides: -3x ≤ 6

Step 2: Divide both sides by -3 (remember to reverse the inequality sign): x ≥ -2

Thus, the solution is x ≥ -2, which means all real numbers greater than or equal to -2.

Interval Notation

Solutions to inequalities are often expressed using interval notation:

• (a, b) represents all numbers between a and b, not including a or b
• [a, b] represents all numbers between a and b, including both a and b
• (a, ∞) represents all numbers greater than a
• (-∞, b) represents all numbers less than b

For example, x > 2 is written as (2, ∞) in interval notation, while x ≥ -2 is written as [-2, ∞).

Applications of Inequalities

Inequalities have numerous real-world applications across various fields:

• Economics: Budget constraints and resource allocation
• Business: Profit calculations and break-even analysis
• Engineering: Tolerance specifications and safety margins
• Physics: Defining ranges of possible values in measurements
• Computer Science: Algorithm analysis and complexity bounds
• Everyday Life: Determining phone plan options, budgeting expenses

Common Mistakes to Avoid

When solving inequalities, students often make these common errors:

• Forgetting to reverse the inequality sign when multiplying or dividing by a negative number
• Incorrectly graphing the solution on a number line
• Mishandling the endpoints when the inequality includes equality (≤ or ≥) versus when it doesn't (< or >)
• Incorrectly translating word problems into inequalities
• Failing to check the solution by testing values in the original inequality

Frequently Asked Questions

Why do we reverse the inequality sign when multiplying or dividing by a negative number?

Consider the true inequality 3 < 5. If we multiply both sides by -1, we get -3 and -5. But -3 is actually greater than -5, so we need to reverse the sign to maintain the truth: -3 > -5. This principle applies to all inequalities when multiplying or dividing by negatives.

What's the difference between solving equations and solving inequalities?

While both processes involve isolating the variable, inequalities have the additional rule about reversing the sign when multiplying/dividing by negatives. Also, solutions to equations are typically specific values, while solutions to inequalities are usually ranges of values.

Can inequalities have no solution?

Yes, some inequalities have no solution. For example, x + 2 < x + 1 simplifies to 2 < 1, which is false for all values of x, meaning no solution exists.

Can inequalities have all real numbers as solutions?

Yes, some inequalities are true for all real numbers. For example, x + 2 > x + 1 simplifies to 2 > 1, which is true for all values of x.

How do I check if my solution to an inequality is correct?

Select test values from your solution interval and from outside it, then substitute these values into the original inequality. Values from your solution should satisfy the inequality, while values outside should not.

What are compound inequalities?

Compound inequalities are two or more inequalities joined by "and" or "or." For "and" inequalities, the solution is the intersection of the individual solutions. For "or" inequalities, the solution is the union of the individual solutions.

How are inequalities used in real life?

Inequalities model many real-world situations where there are limits or constraints, such as budgeting (expenses ≤ income), manufacturing (production ≥ demand), and physics (measurements within tolerance ranges).