Solving Inequalities: Finding The Natural Number 'n'

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Hey math enthusiasts! Today, we're diving into the world of inequalities and figuring out how to find the natural number 'n' that satisfies a bunch of different conditions. Don't worry, it's not as scary as it sounds! We'll break down each problem step by step, making sure everyone understands the process. This is a great exercise for strengthening your algebra skills, and it's also super practical for various real-world scenarios. Ready to get started? Let's go!

Understanding the Basics of Inequalities

Before we jump into the problems, let's quickly recap what inequalities are all about. Inequalities are mathematical statements that compare two expressions, but instead of using an equals sign (=), they use symbols like:

  • < (less than)
  • > (greater than)
  • <= (less than or equal to)
  • >= (greater than or equal to)

Our goal in these problems is to find the values of 'n' that make the inequality true. This often involves isolating 'n' on one side of the inequality. The key thing to remember is that when you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality sign. For example, if you have x > 2 and you multiply both sides by -1, you get -x < -2. Got it? Cool! Let's now delve into the first set of inequalities.

Solving Inequalities: Step-by-Step Solutions

Now, let's solve each inequality one by one to find the possible values of 'n'.

a) n * 5 < 10

  • Original Inequality: n * 5 < 10
  • Step 1: Divide both sides by 5: (n * 5) / 5 < 10 / 5 This simplifies to n < 2.
  • Solution: n can be any natural number less than 2. The natural numbers are 1, 2, 3, .... Therefore, n = 1.

b) 3 * n < 7

  • Original Inequality: 3 * n < 7
  • Step 1: Divide both sides by 3: (3 * n) / 3 < 7 / 3. This simplifies to n < 2.333...
  • Solution: Since 'n' must be a natural number, the possible values for n are 1 and 2. Thus, n = 1, 2.

c) 5(n + 3) < 21

  • Original Inequality: 5(n + 3) < 21
  • Step 1: Distribute the 5: 5n + 15 < 21
  • Step 2: Subtract 15 from both sides: 5n + 15 - 15 < 21 - 15. This simplifies to 5n < 6.
  • Step 3: Divide both sides by 5: 5n / 5 < 6 / 5. This simplifies to n < 1.2.
  • Solution: Since 'n' must be a natural number, the only possible value for n is 1. Therefore, n = 1.

d) (4 - n) * 2 < 11

  • Original Inequality: (4 - n) * 2 < 11
  • Step 1: Distribute the 2: 8 - 2n < 11
  • Step 2: Subtract 8 from both sides: 8 - 2n - 8 < 11 - 8. This simplifies to -2n < 3.
  • Step 3: Divide both sides by -2 (and flip the inequality sign!): -2n / -2 > 3 / -2. This simplifies to n > -1.5.
  • Solution: Since 'n' must be a natural number, the possible values are 1, 2, 3, .... Thus, n = 1, 2, 3, ...

e) n : 3 < 4

  • Original Inequality: n : 3 < 4 (This can also be written as n/3 < 4)
  • Step 1: Multiply both sides by 3: (n / 3) * 3 < 4 * 3. This simplifies to n < 12.
  • Solution: n can be any natural number less than 12. Therefore, n = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11.

f) (3 + 4n) * 12 < 144

  • Original Inequality: (3 + 4n) * 12 < 144
  • Step 1: Divide both sides by 12: (3 + 4n) < 12
  • Step 2: Subtract 3 from both sides: 4n < 9
  • Step 3: Divide both sides by 4: n < 2.25
  • Solution: Since 'n' must be a natural number, the possible values for n are 1 and 2. Therefore, n = 1, 2.

g) (9n + 15) : 3 < 20

  • Original Inequality: (9n + 15) : 3 < 20 (This can also be written as (9n + 15)/3 < 20)
  • Step 1: Multiply both sides by 3: 9n + 15 < 60
  • Step 2: Subtract 15 from both sides: 9n < 45
  • Step 3: Divide both sides by 9: n < 5
  • Solution: n can be any natural number less than 5. Therefore, n = 1, 2, 3, 4.

h) 2 * (n - 2) > 6

  • Original Inequality: 2 * (n - 2) > 6
  • Step 1: Divide both sides by 2: n - 2 > 3
  • Step 2: Add 2 to both sides: n > 5
  • Solution: n can be any natural number greater than 5. Therefore, n = 6, 7, 8, ....

i) 4n + 7 < 5n

  • Original Inequality: 4n + 7 < 5n
  • Step 1: Subtract 4n from both sides: 7 < n
  • Step 2: Rewrite: n > 7
  • Solution: n can be any natural number greater than 7. Therefore, n = 8, 9, 10, ....

Key Takeaways and Tips

So, what did we learn, guys? First off, we've refreshed our understanding of inequalities and how they differ from equations. Secondly, we've honed our skills in solving for a variable (in this case, 'n') within inequalities. Remember these key points:

  • Isolate the Variable: The main goal is to get 'n' by itself on one side of the inequality. Use addition, subtraction, multiplication, and division to achieve this.
  • Flip the Sign (When Necessary): Always remember to flip the inequality sign when multiplying or dividing both sides by a negative number.
  • Consider the Domain: Since we're dealing with natural numbers, make sure your solutions only include positive whole numbers.
  • Practice Makes Perfect: The more you practice solving inequalities, the more comfortable you'll become. Try creating your own problems or find some online to keep your skills sharp.

By following these steps and keeping these tips in mind, you'll be well on your way to mastering inequalities. Keep practicing, and you'll be solving these problems like a pro in no time! Remember, math is all about practice, and understanding the concepts is the most important part of it. Keep up the great work! You've got this!

Conclusion

Well, there you have it, folks! We've successfully solved each inequality and found the possible values for the natural number 'n'. We started with the basics of inequalities, went through the step-by-step solutions of the given problems, and wrapped up with some key takeaways and tips to help you in the future. Remember, with practice and a good understanding of the concepts, you can tackle any inequality problem that comes your way. Keep learning, keep practicing, and most importantly, keep enjoying the world of mathematics!