How to Solve an Inequality: A Comprehensive Guide
Inequalities are mathematical statements comparing two expressions, showing that one is greater than, greater than or equal to, less than, or less than or equal to the other. Solving inequalities involves finding the range of values that satisfy the given inequality. This guide will walk you through the process, covering various types and techniques.
Understanding Inequality Symbols
Before diving into solutions, let's refresh our understanding of the symbols used in inequalities:
- >: Greater than
- ≥: Greater than or equal to
- <: Less than
- ≤: Less than or equal to
These symbols are crucial for interpreting and solving inequalities correctly.
Basic Steps to Solve Inequalities
Solving inequalities is similar to solving equations, but with a few key differences. Here's a general approach:
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Simplify both sides: Combine like terms and simplify expressions on both sides of the inequality.
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Isolate the variable: Use inverse operations (addition, subtraction, multiplication, division) to isolate the variable on one side of the inequality. Remember the crucial rule: When multiplying or dividing both sides by a negative number, you must reverse the inequality sign.
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Check your solution: Substitute a value from your solution set back into the original inequality to verify it satisfies the condition.
Types of Inequalities and Solving Techniques
Let's explore different types of inequalities and how to approach them:
1. Linear Inequalities
These inequalities involve variables raised to the power of 1. They are the simplest form and generally follow the basic steps outlined above.
Example: Solve 2x + 5 > 9
- Subtract 5 from both sides: 2x > 4
- Divide both sides by 2: x > 2
Solution: x > 2 (all values greater than 2 satisfy the inequality)
2. Compound Inequalities
These inequalities combine two or more inequalities using "and" or "or".
- "And" inequalities: The solution must satisfy both inequalities.
- "Or" inequalities: The solution satisfies at least one of the inequalities.
Example (And): Solve 3x - 1 > 5 and x + 2 < 8
- Solve each inequality separately:
- 3x - 1 > 5 => x > 2
- x + 2 < 8 => x < 6
- Find the intersection: The solution is the range where both conditions are true: 2 < x < 6
Example (Or): Solve 2x + 4 ≤ 0 or x - 3 > 1
- Solve each inequality separately:
- 2x + 4 ≤ 0 => x ≤ -2
- x - 3 > 1 => x > 4
- Combine the solutions: The solution includes all values less than or equal to -2 or greater than 4.
3. Quadratic Inequalities
These inequalities involve variables raised to the power of 2. Solving them often requires factoring or using the quadratic formula to find the roots, then testing intervals.
Example: Solve x² - 4x + 3 < 0
- Factor the quadratic: (x - 1)(x - 3) < 0
- Find the roots: x = 1 and x = 3
- Test intervals: Test values in the intervals (-∞, 1), (1, 3), and (3, ∞) to determine which satisfy the inequality. The solution is 1 < x < 3.
Graphing Inequalities
Graphing inequalities visually represents the solution set on a number line. Use open circles for ">" and "<" (values not included) and closed circles for "≥" and "≤" (values included).
Advanced Techniques
More complex inequalities might involve absolute values, rational expressions, or systems of inequalities. These often require a combination of the techniques mentioned above and careful consideration of domains and restrictions.
This comprehensive guide provides a solid foundation for solving various types of inequalities. Remember to practice regularly to master these techniques and confidently tackle any inequality problem you encounter.
