Find All The Possible Values Of { N $}$ Given The Following Conditions:1. { 5n + 3 \ \textless \ 18 $}$2. { \frac{7n}{n^2 + 10} \geq 1 $}$

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Introduction

In mathematics, solving inequalities and rational inequalities is a crucial aspect of algebra and calculus. These types of problems require a deep understanding of mathematical concepts and techniques. In this article, we will focus on finding all possible values of n{ n } given two conditions:

  1. 5n+3<18{ 5n + 3 < 18 }
  2. 7nn2+10β‰₯1{ \frac{7n}{n^2 + 10} \geq 1 }

We will break down each condition, solve the inequalities, and then find the intersection of the solutions to determine the possible values of n{ n }.

Solving the First Inequality: 5n+3<18{ 5n + 3 < 18 }

To solve the first inequality, we need to isolate the variable n{ n }. We can start by subtracting 3 from both sides of the inequality:

5n+3βˆ’3<18βˆ’3{ 5n + 3 - 3 < 18 - 3 }

This simplifies to:

5n<15{ 5n < 15 }

Next, we can divide both sides of the inequality by 5 to solve for n{ n }:

5n5<155{ \frac{5n}{5} < \frac{15}{5} }

This simplifies to:

n<3{ n < 3 }

So, the solution to the first inequality is n<3{ n < 3 }.

Solving the Second Inequality: 7nn2+10β‰₯1{ \frac{7n}{n^2 + 10} \geq 1 }

To solve the second inequality, we need to isolate the variable n{ n }. We can start by multiplying both sides of the inequality by n2+10{ n^2 + 10 } to eliminate the fraction:

7nβ‰₯n2+10{ 7n \geq n^2 + 10 }

Next, we can rearrange the inequality to get a quadratic expression:

n2βˆ’7n+10≀0{ n^2 - 7n + 10 \leq 0 }

We can factor the quadratic expression:

(nβˆ’2)(nβˆ’5)≀0{ (n - 2)(n - 5) \leq 0 }

This inequality is true when 2≀n≀5{ 2 \leq n \leq 5 }.

Finding the Intersection of the Solutions

Now that we have solved both inequalities, we need to find the intersection of the solutions. The solution to the first inequality is n<3{ n < 3 }, and the solution to the second inequality is 2≀n≀5{ 2 \leq n \leq 5 }.

To find the intersection, we need to find the values of n{ n } that satisfy both inequalities. We can see that the intersection of the solutions is 2≀n<3{ 2 \leq n < 3 }.

Conclusion

In this article, we have solved two inequalities and found the intersection of the solutions. The first inequality was 5n+3<18{ 5n + 3 < 18 }, and the solution was n<3{ n < 3 }. The second inequality was 7nn2+10β‰₯1{ \frac{7n}{n^2 + 10} \geq 1 }, and the solution was 2≀n≀5{ 2 \leq n \leq 5 }. The intersection of the solutions was 2≀n<3{ 2 \leq n < 3 }.

Final Answer

The final answer is 2≀n<3{ \boxed{2 \leq n < 3} }.

Step-by-Step Solution

Here is a step-by-step solution to the problem:

  1. Solve the first inequality: 5n+3<18{ 5n + 3 < 18 }
    • Subtract 3 from both sides: 5n<15{ 5n < 15 }
    • Divide both sides by 5: n<3{ n < 3 }
  2. Solve the second inequality: 7nn2+10β‰₯1{ \frac{7n}{n^2 + 10} \geq 1 }
    • Multiply both sides by n2+10{ n^2 + 10 }: 7nβ‰₯n2+10{ 7n \geq n^2 + 10 }
    • Rearrange the inequality: n2βˆ’7n+10≀0{ n^2 - 7n + 10 \leq 0 }
    • Factor the quadratic expression: (nβˆ’2)(nβˆ’5)≀0{ (n - 2)(n - 5) \leq 0 }
    • Solve the inequality: 2≀n≀5{ 2 \leq n \leq 5 }
  3. Find the intersection of the solutions
    • The intersection of the solutions is 2≀n<3{ 2 \leq n < 3 }

Key Concepts

  • Solving linear inequalities
  • Solving quadratic inequalities
  • Finding the intersection of solutions

Common Mistakes

  • Failing to isolate the variable in the inequality
  • Failing to consider the direction of the inequality
  • Failing to find the intersection of the solutions

Real-World Applications

  • Solving inequalities is a crucial aspect of many real-world problems, such as finance, engineering, and economics.
  • Understanding how to solve inequalities can help you make informed decisions and solve complex problems.

Practice Problems

  • Solve the inequality: 2x+5<11{ 2x + 5 < 11 }
  • Solve the inequality: xx2+1β‰₯1{ \frac{x}{x^2 + 1} \geq 1 }
  • Find the intersection of the solutions to the following inequalities: x<2{ x < 2 } and xβ‰₯3{ x \geq 3 }

Q&A: Solving Inequalities and Rational Inequalities

In this article, we will continue to explore the topic of solving inequalities and rational inequalities. We will answer some common questions and provide additional examples to help you understand the concepts.

Q: What is the difference between a linear inequality and a quadratic inequality?

A: A linear inequality is an inequality that can be written in the form ax+b<c{ ax + b < c } or ax+b>c{ ax + b > c }, where a{ a }, b{ b }, and c{ c } are constants. A quadratic inequality, on the other hand, is an inequality that can be written in the form ax2+bx+c<0{ ax^2 + bx + c < 0 } or ax2+bx+c>0{ ax^2 + bx + c > 0 }, where a{ a }, b{ b }, and c{ c } are constants.

Q: How do I solve a rational inequality?

A: To solve a rational inequality, you need to follow these steps:

  1. Multiply both sides of the inequality by the denominator to eliminate the fraction.
  2. Simplify the inequality to get a quadratic expression.
  3. Factor the quadratic expression, if possible.
  4. Use the sign chart method to determine the intervals where the inequality is true.

Q: What is the sign chart method?

A: The sign chart method is a technique used to determine the intervals where a quadratic expression is positive or negative. You create a chart with the x-values on the left and the corresponding signs of the quadratic expression on the right. Then, you use the chart to determine the intervals where the inequality is true.

Q: How do I find the intersection of the solutions to two inequalities?

A: To find the intersection of the solutions to two inequalities, you need to follow these steps:

  1. Solve each inequality separately.
  2. Determine the intervals where each inequality is true.
  3. Find the intersection of the intervals by looking for the values that satisfy both inequalities.

Q: What are some common mistakes to avoid when solving inequalities?

A: Some common mistakes to avoid when solving inequalities include:

  • Failing to isolate the variable in the inequality
  • Failing to consider the direction of the inequality
  • Failing to find the intersection of the solutions
  • Failing to check the solutions in the original inequality

Q: How do I apply the concepts of solving inequalities to real-world problems?

A: The concepts of solving inequalities can be applied to a wide range of real-world problems, including finance, engineering, and economics. For example, you can use inequalities to determine the maximum or minimum value of a function, or to find the optimal solution to a problem.

Q: What are some additional resources for learning about solving inequalities?

A: Some additional resources for learning about solving inequalities include:

  • Online tutorials and videos
  • Textbooks and workbooks
  • Online communities and forums
  • Practice problems and exercises

Conclusion

In this article, we have answered some common questions and provided additional examples to help you understand the concepts of solving inequalities and rational inequalities. We have also discussed some common mistakes to avoid and provided some additional resources for learning about the topic.

Final Answer

The final answer is that solving inequalities and rational inequalities is a crucial aspect of mathematics and can be applied to a wide range of real-world problems.

Step-by-Step Solution

Here is a step-by-step solution to the problem:

  1. Solve the inequality: 2x+5<11{ 2x + 5 < 11 }
    • Subtract 5 from both sides: 2x<6{ 2x < 6 }
    • Divide both sides by 2: x<3{ x < 3 }
  2. Solve the inequality: xx2+1β‰₯1{ \frac{x}{x^2 + 1} \geq 1 }
    • Multiply both sides by x2+1{ x^2 + 1 }: xβ‰₯x2+1{ x \geq x^2 + 1 }
    • Rearrange the inequality: x2βˆ’x+1≀0{ x^2 - x + 1 \leq 0 }
    • Factor the quadratic expression: (xβˆ’1/2)2+3/4≀0{ (x - 1/2)^2 + 3/4 \leq 0 }
    • Solve the inequality: x=1/2{ x = 1/2 }
  3. Find the intersection of the solutions
    • The intersection of the solutions is x=1/2{ x = 1/2 }

Key Concepts

  • Solving linear inequalities
  • Solving quadratic inequalities
  • Finding the intersection of solutions
  • Sign chart method

Common Mistakes

  • Failing to isolate the variable in the inequality
  • Failing to consider the direction of the inequality
  • Failing to find the intersection of the solutions
  • Failing to check the solutions in the original inequality

Real-World Applications

  • Solving inequalities is a crucial aspect of many real-world problems, such as finance, engineering, and economics.
  • Understanding how to solve inequalities can help you make informed decisions and solve complex problems.

Practice Problems

  • Solve the inequality: 3xβˆ’2<5{ 3x - 2 < 5 }
  • Solve the inequality: xx2βˆ’4β‰₯1{ \frac{x}{x^2 - 4} \geq 1 }
  • Find the intersection of the solutions to the following inequalities: x<2{ x < 2 } and xβ‰₯3{ x \geq 3 }