Solve The Inequality:$ \frac{n-2}{5}-\frac{n}{3} \leq 2 }$Choose The Correct Solution A. { N \leq -18$ $ B. { N \geq -18$}$ C. { N \geq -12$}$ D. { N \leq -12$}$

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Introduction

In this article, we will delve into the world of inequalities and learn how to solve them. Inequalities are mathematical expressions that compare two values, and they can be used to describe a wide range of real-world situations. In this case, we will focus on solving the inequality n−25−n3≤2\frac{n-2}{5}-\frac{n}{3} \leq 2. We will break down the solution into manageable steps and provide a clear explanation of each step.

Step 1: Simplify the Inequality

The first step in solving the inequality is to simplify it by combining the fractions. To do this, we need to find a common denominator, which is the least common multiple (LCM) of the denominators. In this case, the LCM of 5 and 3 is 15.

\frac{n-2}{5}-\frac{n}{3} \leq 2
\\
\frac{3(n-2)}{15}-\frac{5n}{15} \leq 2
\\
\frac{3n-6-5n}{15} \leq 2
\\
\frac{-2n-6}{15} \leq 2

Step 2: Multiply Both Sides by 15

To eliminate the fraction, we can multiply both sides of the inequality by 15. This will give us a new inequality with no fractions.

\frac{-2n-6}{15} \leq 2
\\
-2n-6 \leq 30
\\
-2n \leq 36
\\
n \geq -18

Step 3: Check the Solution

Now that we have solved the inequality, we need to check our solution to make sure it is correct. We can do this by plugging in a value of nn that satisfies the inequality and checking if the original inequality is true.

n = -18
\\
\frac{n-2}{5}-\frac{n}{3} \leq 2
\\
\frac{-18-2}{5}-\frac{-18}{3} \leq 2
\\
\frac{-20}{5}+\frac{18}{3} \leq 2
\\
-4+6 \leq 2
\\
2 \leq 2

Conclusion

In this article, we learned how to solve the inequality n−25−n3≤2\frac{n-2}{5}-\frac{n}{3} \leq 2. We broke down the solution into manageable steps and provided a clear explanation of each step. We simplified the inequality, multiplied both sides by 15, and checked our solution to make sure it was correct. The final solution is n≥−18n \geq -18.

Final Answer

The correct solution to the inequality is:

  • A. n≤−18n \leq -18: This is incorrect, as the solution is n≥−18n \geq -18.
  • B. n≥−18n \geq -18: This is correct, as the solution is n≥−18n \geq -18.
  • C. n≥−12n \geq -12: This is incorrect, as the solution is n≥−18n \geq -18.
  • D. n≤−12n \leq -12: This is incorrect, as the solution is n≥−18n \geq -18.

The final answer is B.

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Introduction

In our previous article, we learned how to solve the inequality n−25−n3≤2\frac{n-2}{5}-\frac{n}{3} \leq 2. We broke down the solution into manageable steps and provided a clear explanation of each step. In this article, we will continue to explore the world of inequalities and answer some frequently asked questions.

Q&A

Q: What is an inequality?

A: An inequality is a mathematical expression that compares two values. It can be used to describe a wide range of real-world situations, such as comparing the cost of two products or the time it takes to complete a task.

Q: How do I simplify an inequality?

A: To simplify an inequality, you need to combine the fractions by finding a common denominator. You can then multiply both sides of the inequality by the common denominator to eliminate the fraction.

Q: What is the least common multiple (LCM)?

A: The least common multiple (LCM) is the smallest number that is a multiple of two or more numbers. For example, the LCM of 5 and 3 is 15.

Q: How do I multiply both sides of an inequality by a number?

A: To multiply both sides of an inequality by a number, you need to multiply both the left-hand side and the right-hand side of the inequality by the number. This will give you a new inequality with the same solution as the original inequality.

Q: What is the difference between an inequality and an equation?

A: An inequality is a mathematical expression that compares two values, while an equation is a mathematical expression that states that two values are equal. For example, the inequality x>5x > 5 states that xx is greater than 5, while the equation x=5x = 5 states that xx is equal to 5.

Q: How do I check my solution to an inequality?

A: To check your solution to an inequality, you need to plug in a value of the variable that satisfies the inequality and check if the original inequality is true.

Q: What are some common types of inequalities?

A: Some common types of inequalities include:

  • Linear inequalities: These are inequalities that can be written in the form ax+b≤cax + b \leq c or ax+b≥cax + b \geq c.
  • Quadratic inequalities: These are inequalities that can be written in the form ax2+bx+c≤0ax^2 + bx + c \leq 0 or ax2+bx+c≥0ax^2 + bx + c \geq 0.
  • Rational inequalities: These are inequalities that can be written in the form ax+bcx+d≤0\frac{ax + b}{cx + d} \leq 0 or ax+bcx+d≥0\frac{ax + b}{cx + d} \geq 0.

Conclusion

In this article, we answered some frequently asked questions about solving inequalities. We covered topics such as simplifying inequalities, multiplying both sides of an inequality by a number, and checking solutions to inequalities. We also discussed some common types of inequalities, including linear, quadratic, and rational inequalities.

Final Tips

  • Always read the problem carefully and understand what is being asked.
  • Simplify the inequality by combining fractions and multiplying both sides by the common denominator.
  • Check your solution to the inequality by plugging in a value of the variable that satisfies the inequality.
  • Practice solving inequalities to become more comfortable with the process.

By following these tips and practicing solving inequalities, you will become more confident and proficient in solving inequalities.