N$ Is An Integer.Write The Values Of $n$ Such That $-15 \ \textless \ 3n \leqslant 6$.

Introduction

In mathematics, inequalities are a fundamental concept used to describe relationships between variables. Solving inequalities involves finding the values of the variable that satisfy the given inequality. In this article, we will focus on solving the inequality $-15 \ \textless \ 3n \leqslant 6$ to find the values of the integer $n$ that satisfy the given inequality.

Understanding the Inequality

The given inequality is a compound inequality, which consists of two parts: $-15 \ \textless \ 3n$ and $3n \leqslant 6$. To solve this inequality, we need to isolate the variable $n$ in both parts.

Solving the First Part of the Inequality

The first part of the inequality is $-15 \ \textless \ 3n$. To isolate $n$, we need to divide both sides of the inequality by 3. However, since we are dividing by a negative number, we need to reverse the direction of the inequality.

# Solving the first part of the inequality
# -15 < 3n
# Dividing both sides by 3
# -5 < n

Solving the Second Part of the Inequality

The second part of the inequality is $3n \leqslant 6$. To isolate $n$, we need to divide both sides of the inequality by 3.

# Solving the second part of the inequality
# 3n <= 6
# Dividing both sides by 3
# n <= 2

Combining the Solutions

Now that we have solved both parts of the inequality, we can combine the solutions to find the values of $n$ that satisfy the given inequality. The solution to the first part of the inequality is $n > -5$, and the solution to the second part of the inequality is $n \leqslant 2$. Combining these solutions, we get $-5 < n \leqslant 2$.

Finding the Values of n

Since $n$ is an integer, we need to find the integer values that satisfy the inequality $-5 < n \leqslant 2$. The integer values that satisfy this inequality are $-4, -3, -2, -1, 0, 1,$ and $2$.

Conclusion

In this article, we solved the inequality $-15 \ \textless \ 3n \leqslant 6$ to find the values of the integer $n$ that satisfy the given inequality. We found that the integer values that satisfy the inequality are $-4, -3, -2, -1, 0, 1,$ and $2$. This demonstrates the importance of solving inequalities in mathematics and how they can be used to describe relationships between variables.

Frequently Asked Questions

• What is the solution to the inequality $-15 \ \textless \ 3n \leqslant 6$?
• How do you solve a compound inequality?
• What are the integer values that satisfy the inequality $-5 < n \leqslant 2$?

Final Answer

The final answer is $\boxed{-4, -3, -2, -1, 0, 1, 2}$.

Introduction

In our previous article, we solved the inequality $-15 \ \textless \ 3n \leqslant 6$ to find the values of the integer $n$ that satisfy the given inequality. In this article, we will provide a Q&A guide to help you understand the concept of solving inequalities and how to apply it to different types of inequalities.

Q&A Guide

Q1: What is the solution to the inequality $-15 \ \textless \ 3n \leqslant 6$?

A1: The solution to the inequality $-15 \ \textless \ 3n \leqslant 6$ is $-4, -3, -2, -1, 0, 1,$ and $2$.

Q2: How do you solve a compound inequality?

A2: To solve a compound inequality, you need to isolate the variable in both parts of the inequality. You can do this by adding or subtracting the same value to both sides of the inequality, or by multiplying or dividing both sides of the inequality by the same non-zero value.

Q3: What are the steps to solve an inequality?

A3: The steps to solve an inequality are:

1. Isolate the variable in the inequality.
2. Add or subtract the same value to both sides of the inequality.
3. Multiply or divide both sides of the inequality by the same non-zero value.
4. Check the direction of the inequality.

Q4: How do you check the direction of the inequality?

A4: To check the direction of the inequality, you need to determine whether the inequality is strict (i.e., $<$ or $>$) or non-strict (i.e., $\leq$ or $\geq$). If the inequality is strict, you need to reverse the direction of the inequality when you multiply or divide both sides of the inequality by a negative value.

Q5: What are the different types of inequalities?

A5: There are two main types of inequalities: linear inequalities and quadratic inequalities. Linear inequalities are inequalities that can be written in the form $ax + b \leq c$ or $ax + b \geq c$, where $a$, $b$, and $c$ are constants. Quadratic inequalities are inequalities that can be written in the form $ax^2 + bx + c \leq 0$ or $ax^2 + bx + c \geq 0$, where $a$, $b$, and $c$ are constants.

Q6: How do you solve a quadratic inequality?

A6: To solve a quadratic inequality, you need to factor the quadratic expression and then use the sign of the quadratic expression to determine the solution to the inequality.

Q7: What is the difference between a linear inequality and a quadratic inequality?

A7: The main difference between a linear inequality and a quadratic inequality is the form of the inequality. Linear inequalities are inequalities that can be written in the form $ax + b \leq c$ or $ax + b \geq c$, while quadratic inequalities are inequalities that can be written in the form $ax^2 + bx + c \leq 0$ or $ax^2 + bx + c \geq 0$.

Conclusion

In this article, we provided a Q&A guide to help you understand the concept of solving inequalities and how to apply it to different types of inequalities. We hope that this guide has been helpful in answering your questions and providing you with a better understanding of solving inequalities.

Frequently Asked Questions

• What is the solution to the inequality $-15 \ \textless \ 3n \leqslant 6$?
• How do you solve a compound inequality?
• What are the steps to solve an inequality?
• How do you check the direction of the inequality?
• What are the different types of inequalities?
• How do you solve a quadratic inequality?
• What is the difference between a linear inequality and a quadratic inequality?

Final Answer

The final answer is $\boxed{-4, -3, -2, -1, 0, 1, 2}$.