Solve The Inequality:$\[ -3n + 11 \ \textgreater \ -3(n - 3) + N \\]

Feb 26, 2025 by ADMIN 71 views

Solving the Inequality: A Step-by-Step Guide

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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 $-3n + 11 > -3(n - 3) + n$ . We will break down the solution into manageable steps and provide a clear explanation of each step.

Understanding the Inequality

Before we start solving the inequality, let's take a closer look at what it means. The inequality $-3n + 11 > -3(n - 3) + n$ is a statement that says $-3n + 11$ is greater than $-3(n - 3) + n$ . To solve this inequality, we need to isolate the variable $n$ and determine the values of $n$ that make the inequality true.

Step 1: Simplify the Right-Hand Side

The first step in solving the inequality is to simplify the right-hand side. We can do this by distributing the $-3$ to the terms inside the parentheses.

-3(n - 3) + n = -3n + 9 + n

Now, we can combine like terms to simplify the expression further.

-3n + 9 + n = -2n + 9

So, the inequality becomes:

-3n + 11 > -2n + 9

Step 2: Add 2n to Both Sides

The next step is to add $2n$ to both sides of the inequality. This will help us isolate the variable $n$ .

-3n + 11 + 2n > -2n + 9 + 2n

Simplifying both sides, we get:

-n + 11 > 9

Step 3: Subtract 11 from Both Sides

Now, we need to subtract $11$ from both sides of the inequality. This will help us get closer to isolating the variable $n$ .

-n + 11 - 11 > 9 - 11

Simplifying both sides, we get:

-n > -2

Step 4: Multiply Both Sides by -1

Finally, we need to multiply both sides of the inequality by $-1$ . This will help us isolate the variable $n$ and determine the values of $n$ that make the inequality true.

-n \times -1 > -2 \times -1

Simplifying both sides, we get:

n < 2

Conclusion

In conclusion, we have solved the inequality $-3n + 11 > -3(n - 3) + n$ . We broke down the solution into manageable steps and provided a clear explanation of each step. We simplified the right-hand side, added $2n$ to both sides, subtracted $11$ from both sides, and multiplied both sides by $-1$ to isolate the variable $n$ . The final solution is $n < 2$ , which means that the values of $n$ that make the inequality true are all real numbers less than $2$ .

Example Use Case

The inequality $-3n + 11 > -3(n - 3) + n$ can be used to describe a wide range of real-world situations. For example, suppose we are trying to determine the number of hours that a worker can work in a day without exceeding a certain limit. If the worker can work for $x$ hours without exceeding the limit, then the inequality $-3x + 11 > -3(x - 3) + x$ can be used to determine the maximum number of hours that the worker can work.

Tips and Tricks

When solving inequalities, it's essential to follow the order of operations and to simplify the expression as much as possible. Additionally, it's crucial to be careful when multiplying or dividing both sides of the inequality by a negative number, as this can change the direction of the inequality.

Common Mistakes

When solving inequalities, some common mistakes to avoid include:

Not simplifying the expression enough
Not following the order of operations
Multiplying or dividing both sides of the inequality by a negative number without changing the direction of the inequality

Final Thoughts

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Introduction

In our previous article, we delved into the world of inequalities and learned how to solve them. In this article, we will provide a Q&A guide to help you better understand the concepts and techniques involved in solving inequalities. Whether you are a student, a teacher, or simply someone who wants to improve their math skills, this guide is for you.

Q&A: Solving Inequalities

Q: What is an inequality?

A: An inequality is a mathematical expression that compares two values, and it can be used to describe a wide range of real-world situations.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable and determine the values of the variable that make the inequality true. This can be done by simplifying the expression, adding or subtracting the same value to both sides, and multiplying or dividing both sides by a positive or negative number.

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

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable and determine the values of the variable that make the inequality true. This can be done by simplifying the expression, adding or subtracting the same value to both sides, and multiplying or dividing both sides by a positive or negative number.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you need to factor the quadratic expression, if possible, and then use the sign of the quadratic expression to determine the values of the variable that make the inequality true.

Q: What is the difference between a strict inequality and a non-strict inequality?

A: A strict inequality is an inequality that is written with a strict symbol, such as $>$ or $<$ . A non-strict inequality, on the other hand, is an inequality that is written with a non-strict symbol, such as $\geq$ or $\leq$ .

Q: How do I solve a system of linear inequalities?

A: To solve a system of linear inequalities, you need to find the values of the variables that satisfy all the inequalities in the system. This can be done by graphing the inequalities on a coordinate plane and finding the intersection of the graphs.

Tips and Tricks

When solving inequalities, it's essential to follow the order of operations and to simplify the expression as much as possible.
When multiplying or dividing both sides of the inequality by a negative number, be careful to change the direction of the inequality.
When solving a system of linear inequalities, it's helpful to graph the inequalities on a coordinate plane and find the intersection of the graphs.

Common Mistakes

Not simplifying the expression enough
Not following the order of operations
Multiplying or dividing both sides of the inequality by a negative number without changing the direction of the inequality
Not considering the signs of the variables when solving a system of linear inequalities

Final Thoughts

In conclusion, solving inequalities is a crucial skill in mathematics, and it can be used to describe a wide range of real-world situations. By following the steps outlined in this article and being careful when multiplying or dividing both sides of the inequality by a negative number, you can solve inequalities with confidence. Remember to simplify the expression as much as possible, follow the order of operations, and consider the signs of the variables when solving a system of linear inequalities.

Example Use Case

Resources

Khan Academy: Inequalities
Mathway: Inequalities
Wolfram Alpha: Inequalities