Solve The Inequality: ${ -2w \ \textless \ -18 }$A. { W \ \textgreater \ 9 $}$ B. { W \ \textless \ -16 $}$ C. { W \ \textgreater \ -16 $}$ D. { W \ \textless \ 9 $}$

Mar 13, 2025 by ADMIN 181 views

Solving Inequalities: A Step-by-Step Guide to Understanding and Solving Linear Inequalities

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Introduction

In mathematics, inequalities are a fundamental concept that plays a crucial role in solving various problems in algebra, geometry, and other branches of mathematics. An inequality is a statement that two expressions are not equal, but one is either greater than or less than the other. In this article, we will focus on solving linear inequalities, which are inequalities that can be written in the form of a linear equation. We will use the given inequality ${-2w < -18}$ as an example to demonstrate the steps involved in solving linear inequalities.

Understanding the Basics of Inequalities

Before we dive into solving the given inequality, let's understand the basics of inequalities. An inequality can be written in the form of:

ax + b < c

ax + b > c

where $a$ , $b$ , and $c$ are constants, and $x$ is the variable. The inequality sign can be either less than ( $<$ ) or greater than ( $>$ ).

Solving the Given Inequality

Now, let's solve the given inequality ${-2w < -18}$. To solve this inequality, we need to isolate the variable $w$ on one side of the inequality sign.

Step 1: Divide Both Sides by -2

To isolate the variable $w$ , we need to get rid of the coefficient $-2$ that is being multiplied by $w$ . We can do this by dividing both sides of the inequality by $-2$ . However, when we divide both sides of an inequality by a negative number, we need to reverse the direction of the inequality sign.

\frac{-2w}{-2} > \frac{-18}{-2}

This simplifies to:

w > 9

Step 2: Check the Solution

To check the solution, we can plug in a value of $w$ that is greater than $9$ into the original inequality. Let's say we choose $w = 10$ . Plugging this value into the original inequality, we get:

-2(10) < -18

This simplifies to:

-20 < -18

Since this is true, we can conclude that the solution to the inequality is indeed $w > 9$ .

Conclusion

In conclusion, solving linear inequalities involves isolating the variable on one side of the inequality sign and reversing the direction of the inequality sign when dividing both sides by a negative number. By following the steps outlined in this article, we can solve linear inequalities with ease.

Common Mistakes to Avoid

When solving linear inequalities, there are several common mistakes to avoid. These include:

Not reversing the direction of the inequality sign when dividing both sides by a negative number. This can lead to incorrect solutions.
Not checking the solution. This can lead to incorrect solutions.
Not following the order of operations. This can lead to incorrect solutions.

Tips and Tricks

Here are some tips and tricks to help you solve linear inequalities:

Use a number line. A number line can help you visualize the solution to the inequality.
Use a table. A table can help you organize the solution to the inequality.
Check the solution. This is an important step in solving linear inequalities.

Real-World Applications

Linear inequalities have many real-world applications. Some examples include:

Finance. Linear inequalities can be used to model financial situations, such as determining the minimum amount of money needed to save for a down payment on a house.
Science. Linear inequalities can be used to model scientific situations, such as determining the minimum amount of time needed to complete a experiment.
Engineering. Linear inequalities can be used to model engineering situations, such as determining the minimum amount of material needed to build a bridge.

Final Thoughts

In conclusion, solving linear inequalities is an important skill that has many real-world applications. By following the steps outlined in this article, you can solve linear inequalities with ease. Remember to check the solution and follow the order of operations to avoid common mistakes.

Frequently Asked Questions

Here are some frequently asked questions about solving linear inequalities:

Q: What is a linear inequality? A: A linear inequality is an inequality that can be written in the form of a linear equation.
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 sign and reverse the direction of the inequality sign when dividing both sides by a negative number.
Q: What are some common mistakes to avoid when solving linear inequalities? A: Some common mistakes to avoid when solving linear inequalities include not reversing the direction of the inequality sign when dividing both sides by a negative number, not checking the solution, and not following the order of operations.

References

Here are some references that you can use to learn more about solving linear inequalities:

"Algebra and Trigonometry" by Michael Sullivan. This book provides a comprehensive introduction to algebra and trigonometry, including linear inequalities.
"Mathematics for the Nonmathematician" by Morris Kline. This book provides a comprehensive introduction to mathematics, including linear inequalities.
"Linear Algebra and Its Applications" by Gilbert Strang. This book provides a comprehensive introduction to linear algebra, including linear inequalities.

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Introduction

In our previous article, we discussed the basics of solving linear inequalities. In this article, we will provide a Q&A guide to help you understand and solve linear inequalities. Whether you are a student, a teacher, or a professional, this guide will provide you with the information you need to solve linear inequalities with ease.

Q&A Guide

Q: What is a linear inequality?

A: A linear inequality is an inequality that can be written in the form of a linear equation. It is an inequality that involves a variable and a constant, and can be written in the form of:

ax + b < c

ax + b > c

where $a$ , $b$ , and $c$ 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 sign and reverse the direction of the inequality sign when dividing both sides by a negative number. Here are the steps to follow:

Isolate the variable: Move all the terms with the variable to one side of the inequality sign.
Reverse the direction of the inequality sign: If you divide both sides of the inequality by a negative number, you need to reverse the direction of the inequality sign.
Check the solution: Plug in a value of the variable into the original inequality to check if it is true.

Q: What are some common mistakes to avoid when solving linear inequalities?

A: Some common mistakes to avoid when solving linear inequalities include:

Not reversing the direction of the inequality sign when dividing both sides by a negative number. This can lead to incorrect solutions.
Not checking the solution. This can lead to incorrect solutions.
Not following the order of operations. This can lead to incorrect solutions.

Q: How do I check the solution to a linear inequality?

A: To check the solution to a linear inequality, you need to plug in a value of the variable into the original inequality. If the inequality is true, then the solution is correct. If the inequality is false, then the solution is incorrect.

Q: What are some real-world applications of linear inequalities?

A: Linear inequalities have many real-world applications. Some examples include:

Finance: Linear inequalities can be used to model financial situations, such as determining the minimum amount of money needed to save for a down payment on a house.
Science: Linear inequalities can be used to model scientific situations, such as determining the minimum amount of time needed to complete an experiment.
Engineering: Linear inequalities can be used to model engineering situations, such as determining the minimum amount of material needed to build a bridge.

Q: How do I use a number line to solve a linear inequality?

A: A number line is a graphical representation of the solution to a linear inequality. To use a number line to solve a linear inequality, you need to:

Draw a number line: Draw a number line with the variable on the x-axis.
Mark the solution: Mark the solution to the inequality on the number line.
Check the solution: Check if the solution is correct by plugging in a value of the variable into the original inequality.

Q: How do I use a table to solve a linear inequality?

A: A table is a graphical representation of the solution to a linear inequality. To use a table to solve a linear inequality, you need to:

Create a table: Create a table with the variable on the x-axis and the solution on the y-axis.
Mark the solution: Mark the solution to the inequality on the table.
Check the solution: Check if the solution is correct by plugging in a value of the variable into the original inequality.