To solve inequalities with fractions, the primary strategy is to eliminate the fractions by multiplying both sides of the inequality by the least common multiple (LCM) of the denominators. This simplifies the inequality, making it easier to solve using standard algebraic techniques.
Here's a step-by-step breakdown:
- Identify the Fractions: Note all fractions present in the inequality.
- Find the Least Common Multiple (LCM): Determine the LCM of all the denominators in the inequality.
- Multiply Both Sides by the LCM: Multiply every term on both sides of the inequality by the LCM. This will eliminate the fractions. This creates an equivalent inequality without fractions (as mentioned in the provided video).
- Simplify: After multiplying by the LCM, simplify both sides of the inequality. This usually involves canceling common factors.
- Solve the Inequality: Use standard algebraic techniques (addition, subtraction, multiplication, division) to isolate the variable. Remember: If you multiply or divide both sides of the inequality by a negative number, you must flip the direction of the inequality sign.
- Check the Solution: Choose a value within your solution set and plug it back into the original inequality to ensure it holds true.
Example:
Solve the inequality: (x/2) + 4 > (5/2)
- Fractions: x/2 and 5/2
- LCM: The LCM of 2 and 2 is 2.
- Multiply: Multiply both sides by 2: 2 (x/2 + 4) > 2 (5/2)
- Simplify: x + 8 > 5
- Solve: Subtract 8 from both sides: x > -3
Therefore, the solution is x > -3.
Important Considerations:
- Distribute Carefully: Make sure to distribute the LCM to every term on both sides of the inequality.
- Negative Multiplication/Division: Be extremely careful when multiplying or dividing by a negative number. Remember to flip the inequality sign!
- Compound Inequalities: The same principles apply to compound inequalities involving fractions.
By following these steps, you can effectively solve inequalities containing fractions.
