Table of Contents
Introduction
Inequalities are mathematical expressions involving the symbols > (greater than), < (less than), \(\geq$\)(greater than or equal to), and \(\leq$\)(less than or equal to). Unlike equations, inequalities do not show equality but rather a relation of greater or lesser value. In this document, we will explore how to solve simple inequalities.
Basic Concepts
What are Inequalities?
Inequalities are statements about the relative size or order of two objects. They are used to compare numbers and expressions. For example, \(x > 3$\)means that \(x$\)is greater than 3.
Types of Inequalities
- Greater than: \(x > a\)
- Less than: \(x < a\)
- Greater than or equal to: \(x \geq a\)
- Less than or equal to: \(x \leq a\)
Solving Simple Inequalities
General Rules
- You can add, subtract, multiply, or divide both sides of an inequality by the same positive number without changing the inequality.
- If you multiply or divide both sides by a negative number, reverse the inequality sign.
- Treat the inequality as an equation to find critical values, but remember the solution will be a range.
Examples
# Example 1: Solve x > 5
x <- seq(-10, 10, by = 0.1)
y <- x > 5
plot(x, y, type = 'l', main = "Graph of x > 5")
In this example, the solution is all values of \(x$\)greater than 5. The graph shows this as a line where values turn true after \(x = 5\).
Practice Problems
Solve the following inequalities:
- \(3x + 2 > 8\)
- \(5 - x \leq 2\)
- \(2x - 7 \geq -3\)
- \(-3x + 4 < 1\)
Conclusion
Solving simple inequalities is a fundamental skill in algebra. Understanding and practicing these concepts will help you in more advanced mathematical studies, especially in areas that require analyzing ranges and conditions.