Solve The Inequality N + 7 \textless 5 N − 8 N + 7 \ \textless \ 5n - 8 N + 7 \textless 5 N − 8 .

Mar 8, 2025 by ADMIN 100 views

$Solve the Inequality $n + 7 \ \textless \ 5n - 8$$

Introduction

In mathematics, inequalities are used to compare the values of different expressions. They are an essential part of algebra and are used to solve a wide range of problems. In this article, we will focus on solving the inequality $n + 7 \ \textless \ 5n - 8$ . This inequality involves a single variable, $n$ , and is a linear inequality. We will use algebraic techniques to solve this inequality and find the values of $n$ that satisfy it.

Understanding the Inequality

The given inequality is $n + 7 \ \textless \ 5n - 8$ . This inequality states that the value of $n + 7$ is less than the value of $5n - 8$ . To solve this inequality, we need to isolate the variable $n$ on one side of the inequality.

Isolating the Variable

To isolate the variable $n$ , we need to get all the terms involving $n$ on one side of the inequality. We can do this by subtracting $n$ from both sides of the inequality. This gives us:

$n + 7 - n \ \textless \ 5n - 8 - n$

Simplifying the left-hand side, we get:

$7 \ \textless \ 4n - 8$

Adding 8 to Both Sides

To get rid of the negative term on the right-hand side, we can add 8 to both sides of the inequality. This gives us:

$7 + 8 \ \textless \ 4n - 8 + 8$

Simplifying the left-hand side, we get:

$15 \ \textless \ 4n$

Dividing Both Sides by 4

To isolate the variable $n$ , we need to get rid of the coefficient 4 on the right-hand side. We can do this by dividing both sides of the inequality by 4. This gives us:

$\frac{15}{4} \ \textless \ \frac{4n}{4}$

Simplifying the right-hand side, we get:

$\frac{15}{4} \ \textless \ n$

Conclusion

In this article, we solved the inequality $n + 7 \ \textless \ 5n - 8$ . We used algebraic techniques to isolate the variable $n$ on one side of the inequality. The final solution is $\frac{15}{4} \ \textless \ n$ . This means that the value of $n$ must be greater than $\frac{15}{4}$ to satisfy the inequality.

Example Problems

Here are a few example problems that involve solving linear inequalities:

Solve the inequality $2x + 3 \ \textless \ 5x - 2$ .
Solve the inequality $x - 4 \ \textless \ 3x + 2$ .
Solve the inequality $3x + 2 \ \textless \ 2x - 5$ .

Tips and Tricks

Here are a few tips and tricks that can help you solve linear inequalities:

Always start by isolating the variable on one side of the inequality.
Use algebraic techniques such as addition, subtraction, multiplication, and division to simplify the inequality.
Be careful when dividing both sides of the inequality by a negative number, as this can change the direction of the inequality.
Always check your solution by plugging it back into the original inequality.

Real-World Applications

Linear inequalities have many real-world applications. Here are a few examples:

In finance, linear inequalities can be used to model the growth of an investment over time.
In engineering, linear inequalities can be used to model the behavior of complex systems.
In economics, linear inequalities can be used to model the behavior of markets and economies.

Conclusion

Introduction

Solving linear inequalities can be a challenging task, especially for those who are new to algebra. However, with practice and patience, anyone can master the skills needed to solve these types of inequalities. In this article, we will answer some of the most frequently asked questions about solving linear inequalities.

Q: What is a linear inequality?

A: A linear inequality is an inequality that involves a linear expression. It is an inequality that can be written in the form $ax + b \ \textless \ cx + d$ , where $a$ , $b$ , $c$ , and $d$ are constants, and $x$ is the variable.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable on one side of the inequality. You can do this by adding, subtracting, multiplying, or dividing both sides of the inequality by the same value. The goal is to get all the terms involving the variable on one side of the inequality.

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

A: A linear equation is an equation that involves a linear expression. It is an equation that can be written in the form $ax + b = cx + d$ , where $a$ , $b$ , $c$ , and $d$ are constants, and $x$ is the variable. A linear inequality, on the other hand, is an inequality that involves a linear expression. It is an inequality that can be written in the form $ax + b \ \textless \ cx + d$ , where $a$ , $b$ , $c$ , and $d$ are constants, and $x$ is the variable.

Q: How do I know which direction to flip the inequality sign when I multiply or divide both sides by a negative number?

A: When you multiply or divide both sides of an inequality by a negative number, you need to flip the inequality sign. This is because multiplying or dividing by a negative number changes the direction of the inequality. For example, if you have the inequality $x \ \textless \ 5$ , and you multiply both sides by -1, the inequality becomes $-x \ \textgreater \ -5$ .

Q: Can I use the same methods to solve quadratic inequalities as I do to solve linear inequalities?

A: No, you cannot use the same methods to solve quadratic inequalities as you do to solve linear inequalities. Quadratic inequalities involve quadratic expressions, which are more complex than linear expressions. To solve quadratic inequalities, you need to use more advanced techniques, such as factoring or using the quadratic formula.

Q: How do I know if a solution to a linear inequality is valid?

A: To determine if a solution to a linear inequality is valid, you need to plug the solution back into the original inequality. If the solution satisfies the inequality, then it is a valid solution.

Q: Can I use a calculator to solve linear inequalities?

A: Yes, you can use a calculator to solve linear inequalities. However, you need to be careful when using a calculator, as it may not always give you the correct solution. It's always a good idea to check your solution by plugging it back into the original inequality.

Q: How do I graph a linear inequality on a number line?

A: To graph a linear inequality on a number line, you need to plot a point on the number line that represents the solution to the inequality. If the inequality is of the form $x \ \textless \ a$ , you need to plot a point to the left of $a$ . If the inequality is of the form $x \ \textgreater \ a$ , you need to plot a point to the right of $a$ .

Q: Can I use the same methods to solve systems of linear inequalities as I do to solve systems of linear equations?

A: No, you cannot use the same methods to solve systems of linear inequalities as you do to solve systems of linear equations. Systems of linear inequalities involve multiple inequalities, which require more advanced techniques to solve. To solve systems of linear inequalities, you need to use techniques such as graphing or using linear programming.

Conclusion

Solving linear inequalities can be a challenging task, but with practice and patience, anyone can master the skills needed to solve these types of inequalities. By following the tips and techniques outlined in this article, you can become proficient in solving linear inequalities and apply your skills to a wide range of real-world problems.