Solving inequalities involves isolating the variable, similar to solving equations, but with a few key differences. The goal is to find the range of values that make the inequality true. This process often includes algebraic manipulation.
Steps to Solve Inequalities
The fundamental steps to solve inequalities are:
- Simplify Both Sides: Combine like terms on each side of the inequality.
- Isolate the Variable: Use inverse operations (addition, subtraction, multiplication, and division) to get the variable alone on one side of the inequality.
- Addition and Subtraction: These operations don't change the direction of the inequality.
- Multiplication and Division:
- Multiplying or dividing by a positive number does not change the direction of the inequality.
- Multiplying or dividing by a negative number reverses the direction of the inequality.
- Express the Solution: The solution will be an inequality describing the values of the variable.
Example
Based on the video reference ([Part of a video titled How to Solve Inequalities (NancyPi) - YouTube]()), let’s look at an example:
Suppose you have the inequality: x + 3 ≤ 7
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Isolate the variable x by subtracting 3 from both sides:
- x + 3 - 3 ≤ 7 - 3
- x ≤ 4
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Solution: The solution is x ≤ 4, which means x can be any value less than or equal to 4.
Key Differences from Solving Equations
- Inequality Signs: Instead of an equal sign (=), inequalities use symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to).
- Reversing the Inequality: Remember to reverse the inequality sign when multiplying or dividing both sides by a negative number.
Graphical Representation of Solutions
The solution to an inequality can also be represented graphically on a number line, which is not specifically covered in the reference, but a helpful element.
- Open Circle: Used for < or > to indicate the value is not included in the solution.
- Closed Circle: Used for ≤ or ≥ to indicate the value is included in the solution.
- Shaded Line: Extends to the left or right to show all other values that satisfy the inequality.
Summary
| Step | Description | Example |
|---|---|---|
| 1. Simplify | Combine like terms on each side of the inequality. | 2x + 3 - x < 7 becomes x + 3 < 7 |
| 2. Isolate Variable | Use inverse operations to get the variable alone. Remember to reverse the inequality when multiplying or dividing by a negative number. | x + 3 < 7 becomes x < 4 |
| 3. Express Solution | State the solution clearly using the appropriate inequality sign. | x < 4 |
