The rules for solving inequalities are similar to those for solving equations, with one crucial difference regarding multiplication and division by negative numbers. Here's a breakdown:
Basic Rules for Solving Inequalities
When manipulating inequalities, you aim to isolate the variable on one side. The following operations are generally permitted:
Addition and Subtraction
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Adding the Same Quantity: You can add the same number to both sides of an inequality without changing the inequality sign. For example, if x - 3 < 5, adding 3 to both sides gives x < 8.
- Example:
- Start: a - 2 > 10
- Add 2 to both sides: a - 2 + 2 > 10 + 2
- Result: a > 12
- Example:
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Subtracting the Same Quantity: Similarly, you can subtract the same number from both sides of an inequality without changing the inequality sign. For instance, if x + 2 > 7, subtracting 2 from both sides gives x > 5.
- Example:
- Start: b + 5 < 15
- Subtract 5 from both sides: b + 5 - 5 < 15 - 5
- Result: b < 10
- Example:
Multiplication and Division
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Multiplying or Dividing by a Positive Quantity: When you multiply or divide both sides of an inequality by the same positive number, the inequality sign stays the same. For example, if x/2 < 3, multiplying both sides by 2 gives x < 6.
- Example:
- Start: 3c > 9
- Divide by 3 (positive number) on both sides: 3c / 3 > 9 / 3
- Result: c > 3
- Example:
-
Multiplying or Dividing by a Negative Quantity: This is the key difference. If you multiply or divide both sides of an inequality by the same negative number, you must reverse the inequality sign. For instance, if -x < 4, dividing both sides by -1 gives x > -4 (notice the change from less than to greater than).
- Example:
- Start: -2d < 10
- Divide by -2 (negative number) on both sides and flip the sign: -2d / -2 > 10 / -2
- Result: d > -5
- Example:
Summary Table
| Operation | Rule | Example | Inequality Sign Change |
|---|---|---|---|
| Adding a number | Add the same number to both sides | x - 2 > 5 becomes x > 7 | No |
| Subtracting a number | Subtract the same number from both sides | x + 3 < 10 becomes x < 7 | No |
| Multiplying by a positive number | Multiply both sides by the same positive number | x/2 < 4 becomes x < 8 | No |
| Dividing by a positive number | Divide both sides by the same positive number | 2x > 6 becomes x > 3 | No |
| Multiplying by a negative number | Multiply both sides by the same negative number and reverse the inequality sign | -x < 5 becomes x > -5 | Yes |
| Dividing by a negative number | Divide both sides by the same negative number and reverse the inequality sign | -2x > 8 becomes x < -4 | Yes |
Practical Insights
When solving inequalities, it's always a good idea to:
- Check your answer: Pick a number that satisfies your solution and plug it back into the original inequality to ensure it works.
- Be careful with negatives: Pay close attention to whether you are multiplying or dividing by a negative number because of the sign-reversing rule.
