Solving an inequality is similar to solving an equation, but with a few key differences. Here's a breakdown of how to approach it:
1. Isolate the Variable
Just like with equations, your goal is to get the variable by itself on one side of the inequality.
- Use inverse operations: Add, subtract, multiply, or divide both sides of the inequality by the same number, just like you would with an equation. Remember that multiplying or dividing by a negative number flips the inequality sign (e.g., > becomes <).
Example:
Solve for x in the inequality: 2*x + 5 < 11
- Subtract 5 from both sides: 2*x < 6
- Divide both sides by 2: x < 3
2. Graphing the Solution
Inequalities often have multiple solutions. To visually represent the solution, you can graph it on a number line.
- Open circle: If the inequality uses < or >, use an open circle to indicate that the endpoint is not included in the solution.
- Closed circle: If the inequality uses ≤ or ≥, use a closed circle to indicate that the endpoint is included in the solution.
Example:
The solution x < 3 would be graphed as an open circle at 3, with an arrow pointing to the left, indicating all numbers less than 3.
3. Writing the Solution in Interval Notation
Interval notation is a concise way to express the solution set of an inequality.
- Parentheses: Used for open intervals (e.g., (2, 5) represents all numbers between 2 and 5, not including 2 or 5).
- Brackets: Used for closed intervals (e.g., [2, 5] represents all numbers between 2 and 5, including 2 and 5).
- Infinity: Use ∞ (positive infinity) or -∞ (negative infinity) for unbounded intervals.
Example:
The solution x < 3 can be written in interval notation as (-∞, 3).
4. Compound Inequalities
Sometimes you'll encounter inequalities with two parts, called compound inequalities. These involve the words "and" or "or."
- "And" inequalities: The solution must satisfy both parts of the inequality.
- "Or" inequalities: The solution must satisfy at least one part of the inequality.
Example:
Solve for x in the compound inequality 2x - 1 < 5 and 3x + 2 ≥ 8.
- Solve each part separately:
- 2x - 1 < 5 => x* < 3
- 3x + 2 ≥ 8 => x* ≥ 2
- Combine the solutions: The solution must satisfy both x < 3 and x ≥ 2. This is the interval [2, 3).
5. Real-World Applications
Inequalities are used in various real-world situations, such as:
- Budgeting: Determining how much you can spend on a certain item while staying within your budget.
- Speed limits: Understanding the range of acceptable speeds while driving.
- Temperature ranges: Knowing the safe temperature range for a particular process or environment.
