Solve The Following Inequality:${ 5a \ \textless \ 18 - 4a }$

Mar 9, 2025 by ADMIN 65 views

**Solving Linear Inequalities: A Step-by-Step Guide**

Introduction

Linear inequalities are a fundamental concept in algebra, and solving them is a crucial skill for students to master. In this article, we will focus on solving the linear inequality $5a < 18 - 4a$ . We will break down the solution into manageable steps, using clear and concise language to ensure that readers understand the process.

Understanding Linear Inequalities

A linear inequality is an inequality that can be written in the form $ax < b$ , where $a$ and $b$ are constants, and $x$ is the variable. In this case, the inequality is $5a < 18 - 4a$ . Our goal is to isolate the variable $a$ and determine the values of $a$ that satisfy the inequality.

Step 1: Add 4a to Both Sides

To begin solving the inequality, we need to get all the terms involving $a$ on one side of the inequality. We can do this by adding $4a$ to both sides of the inequality. This gives us:

5a + 4a < 18

Step 2: Combine Like Terms

Now that we have added $4a$ to both sides, we can combine the like terms on the left-hand side of the inequality. In this case, we have $5a + 4a$ , which simplifies to $9a$ . The inequality now becomes:

9a < 18

Step 3: Divide Both Sides by 9

To isolate the variable $a$ , we need to get rid of the coefficient $9$ that is multiplying $a$ . We can do this by dividing both sides of the inequality by $9$ . This gives us:

a < \frac{18}{9}

Step 4: Simplify the Right-Hand Side

The fraction $\frac{18}{9}$ can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is $9$ . This gives us:

a < 2

Conclusion

In this article, we have solved the linear inequality $5a < 18 - 4a$ using a step-by-step approach. We added $4a$ to both sides of the inequality, combined like terms, divided both sides by $9$ , and simplified the right-hand side. The final solution is $a < 2$ , which means that the values of $a$ that satisfy the inequality are all real numbers less than $2$ .

Example Applications

Linear inequalities have many real-world applications, including:

Finance: In finance, linear inequalities are used to model investment portfolios and determine the optimal allocation of assets.
Engineering: In engineering, linear inequalities are used to design and optimize systems, such as electrical circuits and mechanical systems.
Economics: In economics, linear inequalities are used to model economic systems and determine the optimal allocation of resources.

Tips and Tricks

When solving linear inequalities, it's essential to follow the order of operations and to be careful when adding or subtracting terms. Here are some tips and tricks to help you solve linear inequalities:

Use inverse operations: When solving linear inequalities, use inverse operations to isolate the variable. For example, if you have the inequality $2x < 5$ , you can divide both sides by $2$ to get $x < \frac{5}{2}$ .
Check your work: When solving linear inequalities, it's essential to check your work to ensure that the solution is correct. You can do this by plugging in values of the variable into the original inequality and checking if the inequality holds true.
Use visual aids: When solving linear inequalities, it's helpful to use visual aids, such as graphs and charts, to visualize the solution.

Conclusion

Introduction

In our previous article, we discussed how to solve linear inequalities using a step-by-step approach. However, we know that practice makes perfect, and the best way to learn is by asking questions and getting answers. In this article, we will provide a Q&A guide to help you better understand how to solve linear inequalities.

Q: What is a linear inequality?

A: A linear inequality is an inequality that can be written in the form $ax < b$ , where $a$ and $b$ are constants, and $x$ is the variable.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to follow these steps:

Add or subtract the same value to both sides of the inequality to get all the terms involving the variable on one side.
Combine like terms on the left-hand side of the inequality.
Divide both sides of the inequality by the coefficient of the variable to isolate the variable.
Simplify the right-hand 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 can be written in the form $ax = b$ , where $a$ and $b$ are constants, and $x$ is the variable. A linear inequality, on the other hand, is an inequality that can be written in the form $ax < b$ , where $a$ and $b$ are constants, and $x$ is the variable.

Q: How do I know which direction to add or subtract when solving a linear inequality?

A: When solving a linear inequality, you need to add or subtract the same value to both sides of the inequality to get all the terms involving the variable on one side. The direction you add or subtract depends on the inequality sign. If the inequality sign is less than (<), you need to add the same value to both sides. If the inequality sign is greater than (>), you need to subtract the same value from both sides.

Q: What is the significance of the coefficient of the variable in a linear inequality?

A: The coefficient of the variable in a linear inequality is the number that multiplies the variable. When solving a linear inequality, you need to divide both sides of the inequality by the coefficient of the variable to isolate the variable.

Q: How do I check my work when solving a linear inequality?

A: When solving a linear inequality, it's essential to check your work to ensure that the solution is correct. You can do this by plugging in values of the variable into the original inequality and checking if the inequality holds 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 following the order of operations
Not adding or subtracting the same value to both sides of the inequality
Not combining like terms on the left-hand side of the inequality
Not dividing both sides of the inequality by the coefficient of the variable
Not simplifying the right-hand side of the inequality

Q: How can I apply linear inequalities to real-world problems?

A: Linear inequalities have many real-world applications, including:

Finance: In finance, linear inequalities are used to model investment portfolios and determine the optimal allocation of assets.
Engineering: In engineering, linear inequalities are used to design and optimize systems, such as electrical circuits and mechanical systems.
Economics: In economics, linear inequalities are used to model economic systems and determine the optimal allocation of resources.

Conclusion

In conclusion, solving linear inequalities is a crucial skill for students to master. By following the steps outlined in this article and practicing with examples, you can become proficient in solving linear inequalities and apply them to real-world problems. Remember to use inverse operations, check your work, and use visual aids to help you solve linear inequalities. With practice and patience, you will become proficient in solving linear inequalities and be able to apply them to real-world problems.