Solve The Inequality $-28n \leq 7$.

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Introduction

In mathematics, inequalities are a fundamental concept that helps us compare the values of different variables. An inequality is a statement that two expressions are not equal, but one is either greater than, less than, greater than or equal to, or less than or equal to the other. In this article, we will focus on solving the inequality $-28n \leq 7$, which is a linear inequality. We will use algebraic methods to isolate the variable $n$ and find the solution set.

Understanding the Inequality

The given inequality is $-28n \leq 7$. This means that the product of $-28$ and $n$ is less than or equal to $7$. To solve this inequality, we need to isolate the variable $n$.

Isolating the Variable

To isolate the variable $n$, we need to get rid of the coefficient $-28$. We can do this by dividing both sides of the inequality by $-28$. However, when we divide by a negative number, the direction of the inequality sign changes.

Dividing by a Negative Number

When we divide both sides of the inequality by $-28$, we get:

$$n \geq -\frac{7}{28}$$

However, since we divided by a negative number, the direction of the inequality sign changes. This means that the inequality sign should be reversed.

Reversing the Inequality Sign

Since we divided by a negative number, the inequality sign should be reversed. Therefore, the correct inequality is:

$$n \geq -\frac{7}{28}$$

Simplifying the Fraction

The fraction $-\frac{7}{28}$ can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is $7$.

Simplifying the Fraction

$$-\frac{7}{28} = -\frac{1}{4}$$

Writing the Solution Set

The solution set of the inequality $-28n \leq 7$ is all values of $n$ that satisfy the inequality. In this case, the solution set is all values of $n$ that are greater than or equal to $-\frac{1}{4}$.

Writing the Solution Set

$$n \geq -\frac{1}{4}$$

Graphing the Solution Set

The solution set can be graphed on a number line. The number line is divided into two parts: one part represents the values of $n$ that are less than $-\frac{1}{4}$, and the other part represents the values of $n$ that are greater than or equal to $-\frac{1}{4}$.

Graphing the Solution Set

The solution set is represented by the number line:

$$\boxed{-\infty < n \leq -\frac{1}{4}}$$

Conclusion

In this article, we solved the inequality $-28n \leq 7$ using algebraic methods. We isolated the variable $n$ by dividing both sides of the inequality by $-28$, and we reversed the direction of the inequality sign since we divided by a negative number. We simplified the fraction $-\frac{7}{28}$ to $-\frac{1}{4}$ and wrote the solution set as $n \geq -\frac{1}{4}$. We also graphed the solution set on a number line.

Frequently Asked Questions

• What is the solution set of the inequality $-28n \leq 7$?
• How do you isolate the variable $n$ in the inequality $-28n \leq 7$?
• What is the direction of the inequality sign when dividing by a negative number?

Answer to Frequently Asked Questions

• The solution set of the inequality $-28n \leq 7$ is all values of $n$ that are greater than or equal to $-\frac{1}{4}$.
• To isolate the variable $n$ in the inequality $-28n \leq 7$, you need to divide both sides of the inequality by $-28$ and reverse the direction of the inequality sign.
• When dividing by a negative number, the direction of the inequality sign changes.

Final Answer

The final answer is $\boxed{-\infty < n \leq -\frac{1}{4}}$.

Introduction

In mathematics, inequalities are a fundamental concept that helps us compare the values of different variables. An inequality is a statement that two expressions are not equal, but one is either greater than, less than, greater than or equal to, or less than or equal to the other. In this article, we will focus on solving linear inequalities, which are inequalities that can be written in the form $ax \leq b$, where $a$ and $b$ are constants and $x$ is the variable.

Frequently Asked Questions

Q: What is the solution set of the inequality $-28n \leq 7$?

A: The solution set of the inequality $-28n \leq 7$ is all values of $n$ that are greater than or equal to $-\frac{1}{4}$.

Q: How do you isolate the variable $n$ in the inequality $-28n \leq 7$?

A: To isolate the variable $n$ in the inequality $-28n \leq 7$, you need to divide both sides of the inequality by $-28$ and reverse the direction of the inequality sign.

Q: What is the direction of the inequality sign when dividing by a negative number?

A: When dividing by a negative number, the direction of the inequality sign changes.

Q: How do you solve the inequality $2x + 5 \leq 11$?

A: To solve the inequality $2x + 5 \leq 11$, you need to isolate the variable $x$ by subtracting $5$ from both sides of the inequality and then dividing both sides by $2$.

Q: What is the solution set of the inequality $x - 3 \geq 7$?

A: To solve the inequality $x - 3 \geq 7$, you need to isolate the variable $x$ by adding $3$ to both sides of the inequality.

Q: How do you solve the inequality $x + 2 \leq 9$?

A: To solve the inequality $x + 2 \leq 9$, you need to isolate the variable $x$ by subtracting $2$ from both sides of the inequality.

Q: What is the solution set of the inequality $x - 2 \geq 5$?

A: To solve the inequality $x - 2 \geq 5$, you need to isolate the variable $x$ by adding $2$ to both sides of the inequality.

Q: How do you solve the inequality $x + 1 \leq 6$?

A: To solve the inequality $x + 1 \leq 6$, you need to isolate the variable $x$ by subtracting $1$ from both sides of the inequality.

Q: What is the solution set of the inequality $x - 1 \geq 4$?

A: To solve the inequality $x - 1 \geq 4$, you need to isolate the variable $x$ by adding $1$ to both sides of the inequality.

Conclusion

In this article, we have provided a comprehensive guide to solving linear inequalities. We have covered the basics of solving inequalities, including how to isolate the variable, how to reverse the direction of the inequality sign when dividing by a negative number, and how to solve various types of inequalities. We have also provided answers to frequently asked questions, which will help you to better understand the concept of solving inequalities.

Final Answer

The final answer is $\boxed{-\infty < n \leq -\frac{1}{4}}$.