When solving inequalities, it is important to know when to flip the inequality sign. There are specific rules that govern when the inequality sign should be flipped, and understanding these rules is essential for solving inequalities accurately.
Key Facts
- Rule 1: Adding or subtracting the same quantity from both sides of an inequality leaves the inequality symbol unchanged.
- Rule 2: Multiplying or dividing both sides of an inequality by a positive number leaves the inequality symbol unchanged.
- When subtracting a number from both sides of an inequality, you do not need to flip the inequality sign. The sign remains the same.
- However, if you multiply or divide both sides of an inequality by a negative number, you need to flip the inequality sign.
Rule 1: Adding or Subtracting the Same Quantity
When adding or subtracting the same quantity from both sides of an inequality, the inequality symbol remains unchanged. This is because the operation is performed on both sides of the inequality, preserving the relationship between the two sides.
Rule 2: Multiplying or Dividing by a Positive Number
When multiplying or dividing both sides of an inequality by a positive number, the inequality symbol remains unchanged. This is because a positive number does not change the order of the numbers on the number line.
Rule 3: Multiplying or Dividing by a Negative Number
When multiplying or dividing both sides of an inequality by a negative number, the inequality sign must be flipped. This is because a negative number reverses the order of the numbers on the number line.
Subtracting a Number
When subtracting a number from both sides of an inequality, the inequality sign does not need to be flipped. This is because subtracting a number is equivalent to adding its negative, which does not change the inequality symbol according to Rule 1.
Examples
Example 1:
Solve the inequality:
x + 5 > 10
Solution:
Subtract 5 from both sides:
x + 5 - 5 > 10 - 5
x > 5
The inequality sign does not need to be flipped because we are subtracting a number.
Example 2:
Solve the inequality:
-2x < 6
Solution:
Divide both sides by -2 (flipping the inequality sign):
(-2x)/(-2) > 6/(-2)
x > -3
The inequality sign is flipped because we are dividing by a negative number.
Conclusion
Understanding the rules for manipulating inequalities is crucial for solving inequalities correctly. By following these rules, one can ensure that the inequality symbol is flipped when necessary and that the solution is accurate.
FAQs
Do you flip the inequality sign when you subtract a number?
**Answer:** No, you do not flip the inequality sign when you subtract a number. Subtracting a number is equivalent to adding its negative, which does not change the inequality symbol.
When do you flip the inequality sign?
**Answer:** You flip the inequality sign when you multiply or divide both sides of an inequality by a negative number.
Why do you flip the inequality sign when multiplying by a negative number?
**Answer:** Multiplying by a negative number reverses the order of the numbers on the number line. To preserve the inequality relationship, the inequality sign must be flipped.
Is it necessary to flip the inequality sign when dividing by a negative number?
**Answer:** Yes, it is necessary to flip the inequality sign when dividing by a negative number. This is because dividing by a negative number is equivalent to multiplying by its reciprocal, which is a negative number.
What happens if you don’t flip the inequality sign when you should?
**Answer:** If you don’t flip the inequality sign when you should, the solution to the inequality will be incorrect.
Can you give an example of when you would flip the inequality sign?
**Answer:** Yes, if you have the inequality -2x < 6, you would flip the inequality sign when dividing both sides by -2 to solve for x. The new inequality would be x >; -3.
Can you give an example of when you would not flip the inequality sign?
**Answer:** Yes, if you have the inequality x + 5 >; 10, you would not flip the inequality sign when subtracting 5 from both sides. The new inequality would still be x >; 5.
What are the general rules for manipulating inequalities?
**Answer:** The general rules for manipulating inequalities are: