Inequality
Inequality - Solve mathematical problems with step-by-step solutions.
Inequality Calculator
Solve mathematical inequalities and visualize solution sets
Use: x (variable), +, -, *, /, ^, sin, cos, abs, sqrt, etc.
Operators: <, >, <=, >=, !=, ==
Examples
Quick Help
Operators:
< (less), > (greater)
<= (≤), >= (≥)
Functions:
abs, sqrt, sin, cos, exp, log
Example:
2*x - 3 < 7
Understanding Inequalities
Exploring Relationships Beyond Equality.
What are Inequalities?
Inequalities are mathematical statements that compare two values or expressions, showing that one is less than, greater than, or not equal to the other.
Unlike equations, which use an equals sign (=) to show that two things are exactly the same, inequalities use symbols to describe a range of possible values.
They are essential for representing situations where there isn't just one correct answer, but a whole set of possibilities.
Example:Equation: x = 5 (x is exactly 5). Inequality: x > 5 (x can be any number greater than 5).
The Four Main Inequality Symbols
**Greater Than (>):** The value on the left is larger than the value on the right. (e.g., 10 > 3)
**Less Than (<):** The value on the left is smaller than the value on the right. (e.g., 3 < 10)
**Greater Than or Equal To (≥):** The value on the left is larger than or equal to the value on the right. (e.g., x ≥ 5 means x can be 5, 5.1, 6, etc.)
**Less Than or Equal To (≤):** The value on the left is smaller than or equal to the value on the right. (e.g., x ≤ 5 means x can be 5, 4.9, 4, etc.)
Example:Think of the symbol as a crocodile's mouth; it always wants to 'eat' the larger value.
Solving Linear Inequalities
Solving inequalities is very similar to solving equations. The goal is to isolate the variable on one side.
You can add, subtract, multiply, or divide both sides by the same number to maintain the relationship.
**The Golden Rule:** If you multiply or divide both sides of an inequality by a **negative number**, you MUST flip the direction of the inequality sign.
Example:Solving x + 3 < 8 is simple: subtract 3 from both sides to get x < 5.
Case Study: The Flipping Rule
Let's solve the inequality: -2x > 10.
To isolate 'x', we need to divide both sides by -2.
Because we are dividing by a negative number, we must flip the '>' sign to a '<' sign.
(-2x) / -2 < 10 / -2, which simplifies to x < -5.
Example:Original: 6 > 2. Multiply by -1: -6 < -2. The sign must flip to keep the statement true.
Graphing Inequalities on a Number Line
A number line is a great way to visualize the solution to an inequality.
We use a circle on the number and an arrow to show the range of values.
**Open Circle (○):** Used for '>' and '<' to show the number itself is not included in the solution.
**Closed Circle (●):** Used for '≥' and '≤' to show the number is included in the solution.
Example:For x > 2, you would draw an open circle on '2' and an arrow pointing to the right.
Real-World Application: Elevator Capacity
Inequalities are used everywhere. Consider an elevator with a maximum capacity of 2000 lbs.
If 'w' represents the total weight of the passengers, the situation can be described by an inequality: w ≤ 2000.
This tells us that the weight can be exactly 2000 lbs or any value less than that for the elevator to operate safely.
Example:Speed limits (speed ≤ 65 mph) and budgets (spending ≤ $50) are also real-world inequalities.
Key Summary
- Inequalities compare values using >, <, ≥, and ≤.
- Solving inequalities is like solving equations, but with a special rule.
- You MUST flip the inequality sign when multiplying or dividing both sides by a negative number.
- Solutions can be visualized on a number line using open (>, <) or closed (≥, ≤) circles.
Practice Problems
Problem: Solve the inequality: 3x - 5 ≥ 16
First, add 5 to both sides. Then, divide both sides by 3. Since 3 is positive, the sign does not flip.
Solution: 3x ≥ 21 => x ≥ 7
Problem: Solve and graph the inequality: 8 - 4y < 20
Subtract 8 from both sides. Then, divide by -4 and remember to flip the inequality sign.
Solution: -4y < 12 => y > -3. Graph with an open circle on -3 and an arrow to the right.
Problem: You need a score of at least 90 to get an A. Write this as an inequality, where 's' is your score.
'At least' means 90 or more, so we use the 'greater than or equal to' symbol.
Solution: s ≥ 90
Frequently Asked Questions
What is the biggest difference between solving equations and inequalities?
The rule about flipping the inequality sign when multiplying or dividing by a negative number. This is the most crucial difference and the most common source of errors.
Can a variable be on the right side of an inequality?
Yes. '5 < x' is a perfectly valid inequality. It means the same thing as 'x > 5'. It's often easier to read and graph when the variable is on the left, but both are correct.
What about inequalities with variables on both sides?
The strategy is the same. Your first step is to gather the variable terms on one side of the inequality and the constant terms on the other, just as you would with an equation.
Related Math Calculators
Basic Calculator
A simple calculator for basic arithmetic operations including addition, subtraction, multiplication, and division.
Percentage Calculator
Calculate percentages, percentage changes, discounts, and more with our comprehensive percentage calculator.
Scientific Calculator
An advanced calculator with trigonometric, logarithmic, exponential, and memory functions.
Series And Sequence Calculator
Series And Sequence - Solve mathematical problems with step-by-step solutions.
Absolute Value Calculator
Calculate the absolute value of any number or expression.
Algebra Calculator
Solve algebraic equations and expressions with step-by-step solutions.