multi step inequalities worksheet

Guri Bee

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22-10-2024

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Mastering Multi-Step Inequalities: A Comprehensive Guide with Worksheet Examples

Solving inequalities might seem intimidating at first, but with the right approach, it becomes a manageable and even enjoyable skill! This article will guide you through the process of solving multi-step inequalities, offering practical examples and explanations, all based on insights from insightful discussions on GitHub.

What are Multi-Step Inequalities?

Multi-step inequalities are simply inequalities that require more than one step to solve. They involve variables, constants, and operations like addition, subtraction, multiplication, and division. The key is to isolate the variable on one side of the inequality sign while maintaining the inequality's truth.

Solving Multi-Step Inequalities: A Step-by-Step Guide

1. Simplify both sides of the inequality: Combine like terms and distribute any constants or coefficients if necessary. This helps clear the equation and makes it easier to isolate the variable.

   • Example:
     • 2x + 5 > 3x - 2
     • Simplify: 5 > x - 2 (subtracting 2x from both sides)

2. Isolate the variable term: Use inverse operations to get the variable term alone on one side of the inequality. Remember to perform the same operation on both sides to maintain balance.

   • Example:
     • 5 > x - 2
     • Isolate: 7 > x (adding 2 to both sides)

3. Isolate the variable: Use the inverse operation again to isolate the variable completely.

   • Example:
     • 7 > x
     • Isolate: x < 7 (reversing the inequality as we are multiplying by -1)

4. Check your solution: Substitute the solution back into the original inequality to verify that it makes the inequality true.

Practical Examples:

1. Example from GitHub:

   • Problem: 3(x + 2) < 2x + 5
   • Solution:
     • 3x + 6 < 2x + 5 (distribute 3)
     • x + 6 < 5 (subtract 2x from both sides)
     • x < -1 (subtract 6 from both sides)
     • Check: 3(-1 + 2) < 2(-1) + 5 -> 3 < 3. This is not true. Therefore, the solution x < -1 is incorrect. The original inequality has no solution.

2. Real-World Application:

   • Problem: You are trying to save money for a new bike that costs $250. You currently have $50 saved and plan to save $15 each week. How many weeks will it take to save enough for the bike?
   • Inequality: 50 + 15w ≥ 250 (where w is the number of weeks)
   • Solution:
     • 15w ≥ 200 (subtract 50 from both sides)
     • w ≥ 13.33 (divide both sides by 15)
     • Answer: You will need to save for at least 14 weeks to have enough money for the bike.

Multi-Step Inequalities Worksheet

To practice your skills, here is a worksheet inspired by the examples and discussions on GitHub:

1. Solve for x:

   • 4x - 3 > 7x + 6
   • 2(x - 5) ≤ 3x + 1
   • -5x + 2 < 3x - 10

2. Write an inequality for each scenario and solve it:

   • You have a budget of $100 for groceries. You have already spent $30 on meat and produce. How much can you spend on other groceries?
   • You want to buy a new phone that costs $600. You can save $40 each week. How many weeks will it take to save enough for the phone?
   • You are planning a trip that costs $1500. You have $500 saved and can earn $25 per hour. How many hours do you need to work to afford the trip?

Conclusion

Solving multi-step inequalities can be challenging but becomes manageable with practice and a clear understanding of the steps involved. This guide, drawing insights from GitHub discussions and real-world examples, provides a solid foundation for mastering this important concept in algebra. Remember to check your solutions and always keep in mind the context of the problem to ensure you are interpreting the results correctly.