Identify The Properties That Help You Solve This Inequality: $3x \ \textless \ 18$.

Feb 28, 2025 by ADMIN 86 views

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

Introduction

Linear inequalities are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving the inequality $3x < 18$ , which is a simple linear inequality. We will break down the steps involved in solving this inequality and identify the properties that help us arrive at the solution.

Understanding the Inequality

The given inequality is $3x < 18$ . This means that the product of $3$ and $x$ is less than $18$ . To solve this inequality, we need to isolate the variable $x$ .

Step 1: Divide Both Sides by 3

To isolate $x$ , we need to get rid of the coefficient $3$ that is being multiplied by $x$ . We can do this by dividing both sides of the inequality by $3$ . This gives us:

\frac{3x}{3} < \frac{18}{3}

Simplifying both sides, we get:

x < 6

Properties of Inequality

So, what properties of inequality helped us solve this problem? Let's break it down:

Division Property of Inequality: When we divide both sides of an inequality by a positive number, the direction of the inequality remains the same. In this case, we divided both sides by $3$ , which is a positive number.
Multiplication Property of Inequality: When we multiply both sides of an inequality by a positive number, the direction of the inequality remains the same. However, when we multiply both sides by a negative number, the direction of the inequality is reversed.
Addition and Subtraction Property of Inequality: When we add or subtract the same value from both sides of an inequality, the direction of the inequality remains the same.

Real-World Applications

Solving linear inequalities has numerous real-world applications. For example:

Finance: When you are planning a budget, you may need to solve inequalities to determine how much money you can spend on different items.
Science: In scientific experiments, you may need to solve inequalities to determine the range of values for a particular variable.
Engineering: In engineering, you may need to solve inequalities to determine the range of values for a particular variable.

Conclusion

In conclusion, solving linear inequalities is a crucial skill that has numerous real-world applications. By understanding the properties of inequality, we can solve inequalities with ease. In this article, we solved the inequality $3x < 18$ by dividing both sides by $3$ . We also discussed the properties of inequality that helped us arrive at the solution.

Common Mistakes to Avoid

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

Not checking the direction of the inequality: When dividing or multiplying both sides of an inequality, make sure to check the direction of the inequality.
Not simplifying the inequality: Make sure to simplify the inequality after each step to avoid confusion.
Not using the correct properties of inequality: Make sure to use the correct properties of inequality to solve the inequality.

Practice Problems

To practice solving linear inequalities, try the following problems:

$2x < 10$
$x > 5$
$3x - 2 < 12$

Solutions

Here are the solutions to the practice problems:

$2x < 10$ : Divide both sides by $2$ to get $x < 5$ .
$x > 5$ : This inequality is already in its simplest form.
$3x - 2 < 12$ : Add $2$ to both sides to get $3x < 14$ . Then, divide both sides by $3$ to get $x < \frac{14}{3}$ .

Conclusion

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 isolate the variable $x$ . You can do this by adding or subtracting the same value from both sides of the inequality, or by multiplying or dividing both sides by a positive or negative number.

Q: What are the properties of inequality?

A: The properties of inequality are:

Division Property of Inequality: When you divide both sides of an inequality by a positive number, the direction of the inequality remains the same.
Multiplication Property of Inequality: When you multiply both sides of an inequality by a positive number, the direction of the inequality remains the same. However, when you multiply both sides by a negative number, the direction of the inequality is reversed.
Addition and Subtraction Property of Inequality: When you add or subtract the same value from both sides of an inequality, the direction of the inequality remains the same.

Q: How do I check if my solution is correct?

A: To check if your solution is correct, you need to plug the solution back into the original inequality. If the solution satisfies the inequality, then it is correct.

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

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

Not checking the direction of the inequality: When dividing or multiplying both sides of an inequality, make sure to check the direction of the inequality.
Not simplifying the inequality: Make sure to simplify the inequality after each step to avoid confusion.
Not using the correct properties of inequality: Make sure to use the correct properties of inequality to solve the inequality.

Q: How do I solve a linear inequality with fractions?

A: To solve a linear inequality with fractions, you need to get rid of the fractions by multiplying both sides of the inequality by the least common multiple (LCM) of the denominators.

Q: How do I solve a linear inequality with decimals?

A: To solve a linear inequality with decimals, you need to get rid of the decimals by multiplying both sides of the inequality by a power of 10.

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

A: Yes, you can use a calculator to solve linear inequalities. However, make sure to check your solution by plugging it back into the original inequality.

Q: How do I graph a linear inequality?

A: To graph a linear inequality, you need to graph the related linear equation and then shade the region that satisfies the inequality.

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

A: Some real-world applications of linear inequalities include:

Finance: When you are planning a budget, you may need to solve inequalities to determine how much money you can spend on different items.
Science: In scientific experiments, you may need to solve inequalities to determine the range of values for a particular variable.
Engineering: In engineering, you may need to solve inequalities to determine the range of values for a particular variable.

Conclusion

In conclusion, solving linear inequalities is a crucial skill that has numerous real-world applications. By understanding the properties of inequality, we can solve inequalities with ease. In this article, we answered some frequently asked questions about solving linear inequalities and provided some tips and tricks for solving them.